Additive Operator-Difference Schemes: Splitting Schemes 9783110321463, 9783110321432

Table of contents :
Preface
Notation
1 Introduction
1.1 Numerical methods
1.2 Additive operator-difference schemes
1.3 The main results
1.4 Contents of the book
2 Stability of operator-difference schemes
2.1 The Cauchy problem for an operator-differential equation
2.1.1 Hilbert spaces
2.1.2 Linear operators in a finite-dimensional space
2.1.3 Operators in a finite-dimensional Hilbert space
2.1.4 The Cauchy problem for an evolutionary equation of first order
2.1.5 Systems of linear ordinary differential equations
2.1.6 A boundary value problem for a one-dimensional parabolic equation
2.1.7 Equations of second order
2.2 Two-level schemes
2.2.1 Key concepts
2.2.2 Stability with respect to the initial data
2.2.3 Stability with respect to the right-hand side
2.2.4 Schemes with weights
2.3 Three-level schemes
2.3.1 Stability with respect to the initial data
2.3.2 Reduction to a two-level scheme
2.3.3 P-stability of three-level schemes
2.3.4 Estimates in simpler norms
2.3.5 Stability with respect to the right-hand side
2.3.6 Schemes with weights for equations of first order
2.3.7 Schemes with weights for equations of second order
2.4 Stability in finite-dimensional Banach spaces
2.4.1 The Cauchy problem for a system of ordinary differential equations
2.4.2 Scheme with weights
2.4.3 Difference schemes for a one-dimensional parabolic equation
2.5 Stability of projection-difference schemes
2.5.1 Preliminary observations
2.5.2 Stability of finite element techniques
2.5.3 Stability of projection-difference schemes
2.5.4 Conditions for -stability of projection-difference schemes
2.5.5 Schemes with weights
2.5.6 Stability with respect to the right-hand side
2.5.7 Stability of three-level schemes with respect to the initial data
2.5.8 Stability with respect to the right-hand side
2.5.9 Schemes for an equation of first order
3 Operator splitting
3.1 Time-dependent problems of convection-diffusion
3.1.1 Differential problem
3.1.2 Semi-discrete problem
3.1.3 Two-level schemes
3.2 Splitting operators in convection-diffusion problems
3.2.1 Splitting with respect to spatial variables
3.2.2 Splitting with respect to physical processes
3.2.3 Schemes for problems with an operator semibounded from below
3.3 Domain decomposition methods
3.3.1 Preliminaries
3.3.2 Model boundary value problems
3.3.3 Standard finite difference approximations
3.3.4 Domain decomposition
3.3.5 Problems with non-self-adjoint operators
3.4 Difference schemes for time-dependent vector problems
3.4.1 Preliminary discussions
3.4.2 Statement of the problem
3.4.3 Estimates for the solution of differential problems
3.4.4 Approximation in space
3.4.5 Schemes with weights
3.4.6 Alternating triangle method
3.5 Problems of hydrodynamics of an incompressible fluid
3.5.1 Differential problem
3.5.2 Discretization in space
3.5.3 Peculiarities of hydrodynamic equations written in the primitive variables
3.5.4 A priori estimate for the differential problem
3.5.5 Approximation in space
3.5.6 Additive difference schemes
4 Additive schemes of two-component splitting
4.1 Alternating direction implicit schemes
4.1.1 Problem formulation
4.1.2 The Peaceman–Rachford scheme
4.1.3 Stability of the Peaceman–Rachford scheme
4.1.4 Accuracy of the Peaceman–Rachford scheme
4.1.5 Another ADI scheme
4.2 Factorized schemes
4.2.1 General considerations
4.2.2 ADI methods as factorized schemes
4.2.3 Stability and accuracy of factorized schemes
4.2.4 Regularization principle for constructing factorized schemes
4.2.5 Factorized schemes of multicomponent splitting
4.3 Alternating triangle method
4.3.1 General description of the alternating triangle method
4.3.2 Investigation of stability and convergence
4.3.3 Three-level additive schemes
4.3.4 Problems with non-self-adjoint operators
4.4 Equations of second order
4.4.1 Model problem
4.4.2 Factorized schemes
4.4.3 Schemes of the alternating triangle method
5 Schemes of summarized approximation
5.1 Additive formulations of differential problems
5.1.1 Model problem
5.1.2 Intermediate problems
5.1.3 Summarized approximation concept
5.1.4 Schemes of the second-order summarized approximation
5.2 Investigation of schemes of summarized approximation
5.2.1 Schemes of componentwise splitting
5.2.2 Estimates for the intermediate problem solutions
5.2.3 Stability of componentwise splitting schemes
5.2.4 Convergence of componentwise splitting schemes
5.2.5 Convergence of additive schemes in Banach spaces
5.3 Additively averaged schemes
5.3.1 Differential problem
5.3.2 Additive schemes
5.3.3 Stability of additively averaged schemes
5.4 Other variants of componentwise splitting schemes
5.4.1 Fully implicit additive schemes
5.4.2 ADI methods as additive schemes
5.4.3 Additive schemes with second-order accuracy
5.4.4 Convergence of higher-order schemes
6 Regularized additive schemes
6.1 Multiplicative regularization of difference schemes
6.1.1 Regularization principle for difference schemes
6.1.2 Additive regularization
6.1.3 Multiplicative regularization
6.2 Multiplicative regularization of additive schemes
6.2.1 The Cauchy problem for a first-order equation
6.2.2 Regularization of additive schemes
6.2.3 Stability and convergence
6.2.4 Regularized and additively averaged schemes
6.3 Schemes of higher-order accuracy
6.3.1 Statement of the problem
6.3.2 Explicit three-level scheme
6.3.3 Regularized schemes
6.3.4 Additively averaged scheme
6.4 Regularized schemes for equations of second order
6.4.1 Model problem
6.4.2 Regularized scheme
6.4.3 Additively averaged schemes for equations of second order
6.5 Regularized schemes with general regularizers
6.5.1 General regularizers
6.5.2 Additive schemes with a general-form regularizer
6.5.3 Factorized additive schemes
6.5.4 Generalizations
7 Schemes based on approximations of a transition operator
7.1 Operator-difference schemes
7.1.1 Operator-differential problem
7.1.2 Difference approximations in time
7.1.3 SM-stable schemes for problems with a self-adjoint operator
7.1.4 Factorized SM-stable two-level schemes
7.1.5 Problems with a skew-symmetric operator
7.2 Additive schemes with a multiplicative transition operator
7.2.1 Operator-differential problems
7.2.2 Componentwise splitting schemes
7.3 Splitting schemes with an additive transition operator
7.3.1 Additive approximation of a transition operator
7.3.2 Additive schemes
7.3.3 Regularized additive schemes
7.4 Further additive schemes
7.4.1 Schemes of the second order
7.4.2 Factorized schemes
7.4.3 Inhomogeneous approximation of a transition operator
7.4.4 Schemes of higher-order approximation
8 Vector additive schemes
8.1 Vector schemes for first-order equations
8.1.1 Vector differential problem
8.1.2 Stability of vector additive schemes
8.1.3 Stability with respect to the right-hand side
8.2 Stability of vector additive schemes in Banach spaces
8.2.1 Problem formulation
8.2.2 Vector additive scheme
8.2.3 Study on stability
8.3 Schemes of second-order accuracy
8.3.1 Statement of the problem
8.3.2 Three-level vector schemes
8.3.3 Schemes of the alternating triangle method
8.4 Vector schemes for equations of second order
8.4.1 The Cauchy problem for a second-order equation
8.4.2 Vector problem
8.4.3 Scheme with weights
8.4.4 Additive schemes
8.4.5 Stability of additive schemes
9 Iterative methods
9.1 Basics of iterative methods
9.1.1 Problem formulation
9.1.2 Simple iteration method
9.1.3 The Chebyshev iterative method
9.1.4 Two-level variation-type methods
9.1.5 Conjugate gradient method
9.2 Iterative alternating direction method
9.2.1 Iterative method with two-component splitting
9.2.2 Convergence study
9.2.3 Modified iterative method of alternating directions
9.2.4 Multicomponent splitting
9.3 Iterative alternating triangle method
9.3.1 Iterative method
9.3.2 Convergence rate
9.3.3 Modified iterative method of alternating triangles
9.4 Iterative cluster aggregation methods
9.4.1 Transition to a system of equations
9.4.2 Iterative method
9.4.3 Parallel variant
9.4.4 Aggregation of unknowns
10 Splitting of the operator at the time derivative
10.1 Schemes with splitting of the operator at the time derivative
10.1.1 Preliminary discussions
10.1.2 Statement of the problem
10.1.3 Vector problem
10.1.4 Vector additive schemes
10.1.5 Generalizations
10.2 General splitting
10.2.1 Preliminary discussions
10.2.2 Problem formulation
10.2.3 Scheme with weights
10.2.4 Schemes with a diagonal operator
10.2.5 The more general problem
10.3 Explicit-implicit splitting schemes
10.3.1 Introduction
10.3.2 Boundary value problems for systems of equations
10.3.3 Schemes with a diagonal operator
10.3.4 General case
11 Equations with pairwise adjoint operators
11.1 Splitting schemes for a system of equations
11.1.1 Preliminary discussions
11.1.2 Statement of the problem
11.1.3 A priori estimates
11.1.4 Schemes with weights
11.1.5 Splitting schemes to find the p-th component of the solution
11.1.6 Additive schemes for systems of equations
11.2 Additive schemes for a system of first-order equations
11.2.1 Statement of the problem
11.2.2 Examples
11.2.3 Schemes with weights
11.2.4 Explicit-implicit schemes
11.2.5 Additive schemes of componentwise splitting
11.2.6 Regularized additive schemes
11.3 Another class of systems of first-order equations
11.3.1 Problem formulation
11.3.2 Scheme with weights
11.3.3 Additive schemes
11.3.4 More general problems
11.3.5 Problems of hydrodynamics
Bibliography
Index

Citation preview

Petr N. Vabishchevich Additive Operator-Difference Schemes

Petr N. Vabishchevich

Additive OperatorDifference Schemes Splitting Schemes

De Gruyter

Mathematics Subject Classification 2010: 65J08, 65J10, 65M06, 65M12, 65M22, 65M55, 65Z05

ISBN 978-3-11-032143-2 e-ISBN 978-3-11-032146-3 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Typesetting: P T P-Berlin Protago-TEX-Production GmbH, www.ptp-berlin.de Printing and binding: CPI buch bücher.de GmbH, Birkach Printed on acid-free paper Printed in Germany www.degruyter.com

Preface

Applied mathematical modeling is basically concerned with the necessity to solve unsteady problems of mathematical physics. A mathematical model for simulation may include as elements both initial value problems for systems of ODEs and, which is most often, time-dependent PDEs. To construct discretization in space, finite difference schemes or finite element procedures are widely used in various ways. This results in transient problems for systems of ODEs. A specific feature of these problems of mathematical physics is in their high stiffness. In this book, we study mathematical modeling problems in the corresponding finitedimensional real Hilbert or Banach spaces as problems with the initial conditions for operator-differential equations. We investigate linear problems that are written in the form of evolutionary equations of first or second order and their systems. As a rule, these mathematical models are essentially nonlinear – the world is nonlinear, and, as academician Samarskii said, linear models comprise only a particular and very simple case. The primary linear models provide the basis for developing efficient computational algorithms, i.e., for designing elegant theoretical constructions that are used to verify their well-posedness and accuracy. Numerical methods for solving linear problems give us a methodological basis to construct algorithms for nonlinear problems. Discretization in time is conducted using one or another difference approximation. This allows us to move from the Cauchy problem for operator-differential equations to operator-difference schemes. Unconditionally stable schemes are constructed employing implicit schemes. In view of stiffness of ODEs, explicit schemes have no practical interest. Optimization of computational algorithms for solving unsteady problems is associated with simplifications of the problem at the upper time level. A typical situation is the case where the operators of the problem under consideration are represented as the sum of operator terms. Additive operator-difference schemes are attributed to the transition from a complex problem to a chain of simpler problems for the individual terms in this operator splitting. This splitting may have a different nature: the individual operators, e.g., may be associated with splitting with respect to the spatial variables or the decomposed parts may have different treatments in the sense of physical phenomena. The classical examples of additive difference schemes are the ADI algorithms as well as locally one-dimensional schemes. They have been widely used in computational practice for more than half a century. Their study is based on the fundamental concept of summarized approximation. Nowadays, new classes of additive difference schemes are being developed. A major contribution to this research area is provided by the Russian (Soviet) school of computational mathematics.

vi

Preface

The key results obtained in the theory and practical usage of splitting schemes are presented in detail in the book by Marchuk Methods of Splitting, 1989 (in Russian). In 1990 this book was published in English (Handbook of Numerical Analysis, Vol.1. Splitting and Alternating Direction Methods, Elsevier). New research results on the theory of additive schemes (schemes of splitting) are reflected in our joint book Additive Schemes for Problems of Mathematical Physics written with Samarskii. This book was published by Nauka, Moscow in 1999, in Russian, with a small edition. Unfortunately, it has actually gone unnoticed by English-speaking readers. This fact as well as the necessity to reflect the recent progress in constructing and studying additive schemes became the main reason for writing the new book. The book is fundamentally concerned with constructing additive difference schemes to solve numerically unsteady multi-dimensional problems for PDEs. Two classes of schemes are highlighted: methods of splitting with respect to spatial variables (alternating direction methods) and schemes of splitting into physical processes. Regionally additive schemes (domain decomposition methods) are also developed for parallel computing. Unconditionally stable additive schemes of multicomponent splitting are considered for evolutionary equations of first and second order as well as for systems of equations. The matter of the book is primarily based on the results derived by the author and his co-authors during the last twenty years. To present the material, we use the minimal mathematical tools concerned with the basic properties of operators in finite-dimensional spaces. The study of additive schemes is based on the general theory of stability for operator-difference schemes developed by Samarskii in the framework of finite-dimensional Hilbert spaces.

Moscow, April, 2013

Petr N. Vabishchevich

Contents

Preface

v

Notation

xv

1 Introduction

1

1.1 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 Additive operator-difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.4 Contents of the book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2 Stability of operator-difference schemes

14

2.1 The Cauchy problem for an operator-differential equation . . . . . . . . . . 2.1.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Linear operators in a finite-dimensional space . . . . . . . . . . . . . 2.1.3 Operators in a finite-dimensional Hilbert space . . . . . . . . . . . . 2.1.4 The Cauchy problem for an evolutionary equation of first order 2.1.5 Systems of linear ordinary differential equations . . . . . . . . . . . 2.1.6 A boundary value problem for a one-dimensional parabolic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.7 Equations of second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 14 16 17 19 20

2.2 Two-level schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Key concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Stability with respect to the initial data . . . . . . . . . . . . . . . . . . . 2.2.3 Stability with respect to the right-hand side . . . . . . . . . . . . . . . 2.2.4 Schemes with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24 24 26 29 31

2.3 Three-level schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Stability with respect to the initial data . . . . . . . . . . . . . . . . . . . 2.3.2 Reduction to a two-level scheme . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 -stability of three-level schemes . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Estimates in simpler norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Stability with respect to the right-hand side . . . . . . . . . . . . . . . 2.3.6 Schemes with weights for equations of first order . . . . . . . . . . 2.3.7 Schemes with weights for equations of second order . . . . . . . .

32 32 34 36 38 40 40 42

21 23

viii

3

Contents

2.4 Stability in finite-dimensional Banach spaces . . . . . . . . . . . . . . . . . . . . 2.4.1 The Cauchy problem for a system of ordinary differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Scheme with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Difference schemes for a one-dimensional parabolic equation .

43 43 45 47

2.5 Stability of projection-difference schemes . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Preliminary observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Stability of finite element techniques . . . . . . . . . . . . . . . . . . . . 2.5.3 Stability of projection-difference schemes . . . . . . . . . . . . . . . . 2.5.4 Conditions for -stability of projection-difference schemes . . . 2.5.5 Schemes with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.6 Stability with respect to the right-hand side . . . . . . . . . . . . . . . 2.5.7 Stability of three-level schemes with respect to the initial data 2.5.8 Stability with respect to the right-hand side . . . . . . . . . . . . . . . 2.5.9 Schemes for an equation of first order . . . . . . . . . . . . . . . . . . .

47 48 49 51 53 55 57 59 60 61

Operator splitting

63

3.1 Time-dependent problems of convection-diffusion . . . . . . . . . . . . . . . . 3.1.1 Differential problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Semi-discrete problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Two-level schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63 63 68 70

3.2 Splitting operators in convection-diffusion problems . . . . . . . . . . . . . . 3.2.1 Splitting with respect to spatial variables . . . . . . . . . . . . . . . . . 3.2.2 Splitting with respect to physical processes . . . . . . . . . . . . . . . 3.2.3 Schemes for problems with an operator semibounded from below . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77 77 78 80

3.3 Domain decomposition methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Model boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Standard finite difference approximations . . . . . . . . . . . . . . . . 3.3.4 Domain decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Problems with non-self-adjoint operators . . . . . . . . . . . . . . . . .

82 82 85 87 91 98

3.4 Difference schemes for time-dependent vector problems . . . . . . . . . . . 3.4.1 Preliminary discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Estimates for the solution of differential problems . . . . . . . . . . 3.4.4 Approximation in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Schemes with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.6 Alternating triangle method . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 102 104 106 108 109

Contents

3.5 Problems of hydrodynamics of an incompressible fluid . . . . . . . . . . . . 3.5.1 Differential problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Discretization in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Peculiarities of hydrodynamic equations written in the primitive variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 A priori estimate for the differential problem . . . . . . . . . . . . . . 3.5.5 Approximation in space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.6 Additive difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Additive schemes of two-component splitting

ix 112 112 114 117 118 119 121 123

4.1 Alternating direction implicit schemes . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 The Peaceman–Rachford scheme . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Stability of the Peaceman–Rachford scheme . . . . . . . . . . . . . . 4.1.4 Accuracy of the Peaceman–Rachford scheme . . . . . . . . . . . . . 4.1.5 Another ADI scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 123 124 125 126 127

4.2 Factorized schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 General considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 ADI methods as factorized schemes . . . . . . . . . . . . . . . . . . . . . 4.2.3 Stability and accuracy of factorized schemes . . . . . . . . . . . . . . 4.2.4 Regularization principle for constructing factorized schemes . 4.2.5 Factorized schemes of multicomponent splitting . . . . . . . . . . .

127 128 128 129 131 133

4.3 Alternating triangle method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 General description of the alternating triangle method . . . . . . . 4.3.2 Investigation of stability and convergence . . . . . . . . . . . . . . . . 4.3.3 Three-level additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.4 Problems with non-self-adjoint operators . . . . . . . . . . . . . . . . .

134 135 136 137 139

4.4 Equations of second order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Factorized schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Schemes of the alternating triangle method . . . . . . . . . . . . . . .

140 141 142 143

5 Schemes of summarized approximation 5.1 Additive formulations of differential problems . . . . . . . . . . . . . . . . . . . 5.1.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Intermediate problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Summarized approximation concept . . . . . . . . . . . . . . . . . . . . . 5.1.4 Schemes of the second-order summarized approximation . . . .

144 144 144 145 147 148

5.2 Investigation of schemes of summarized approximation . . . . . . . . . . . 150 5.2.1 Schemes of componentwise splitting . . . . . . . . . . . . . . . . . . . . 150 5.2.2 Estimates for the intermediate problem solutions . . . . . . . . . . . 151

x

Contents

5.2.3 5.2.4 5.2.5

6

Stability of componentwise splitting schemes . . . . . . . . . . . . . 153 Convergence of componentwise splitting schemes . . . . . . . . . . 154 Convergence of additive schemes in Banach spaces . . . . . . . . . 155

5.3 Additively averaged schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Differential problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Stability of additively averaged schemes . . . . . . . . . . . . . . . . .

156 156 157 158

5.4 Other variants of componentwise splitting schemes . . . . . . . . . . . . . . . 5.4.1 Fully implicit additive schemes . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 ADI methods as additive schemes . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Additive schemes with second-order accuracy . . . . . . . . . . . . . 5.4.4 Convergence of higher-order schemes . . . . . . . . . . . . . . . . . . .

160 160 161 162 163

Regularized additive schemes

167

6.1 Multiplicative regularization of difference schemes . . . . . . . . . . . . . . . 6.1.1 Regularization principle for difference schemes . . . . . . . . . . . . 6.1.2 Additive regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Multiplicative regularization . . . . . . . . . . . . . . . . . . . . . . . . . . .

167 167 168 170

6.2 Multiplicative regularization of additive schemes . . . . . . . . . . . . . . . . . 6.2.1 The Cauchy problem for a first-order equation . . . . . . . . . . . . . 6.2.2 Regularization of additive schemes . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Stability and convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Regularized and additively averaged schemes . . . . . . . . . . . . .

171 171 172 173 175

6.3 Schemes of higher-order accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Explicit three-level scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Regularized schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Additively averaged scheme . . . . . . . . . . . . . . . . . . . . . . . . . . .

176 176 177 178 179

6.4 Regularized schemes for equations of second order . . . . . . . . . . . . . . . 6.4.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Regularized scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Additively averaged schemes for equations of second order . .

180 180 181 182

6.5 Regularized schemes with general regularizers . . . . . . . . . . . . . . . . . . . 6.5.1 General regularizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Additive schemes with a general-form regularizer . . . . . . . . . . 6.5.3 Factorized additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183 183 185 186 187

Contents

7 Schemes based on approximations of a transition operator 7.1 Operator-difference schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Operator-differential problem . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Difference approximations in time . . . . . . . . . . . . . . . . . . . . . . 7.1.3 SM-stable schemes for problems with a self-adjoint operator . 7.1.4 Factorized SM-stable two-level schemes . . . . . . . . . . . . . . . . . 7.1.5 Problems with a skew-symmetric operator . . . . . . . . . . . . . . . .

xi 190 190 190 191 194 199 203

7.2 Additive schemes with a multiplicative transition operator . . . . . . . . . 204 7.2.1 Operator-differential problems . . . . . . . . . . . . . . . . . . . . . . . . . 204 7.2.2 Componentwise splitting schemes . . . . . . . . . . . . . . . . . . . . . . 206 7.3 Splitting schemes with an additive transition operator . . . . . . . . . . . . . 7.3.1 Additive approximation of a transition operator . . . . . . . . . . . . 7.3.2 Additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Regularized additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . .

208 209 209 211

7.4 Further additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Schemes of the second order . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Factorized schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Inhomogeneous approximation of a transition operator . . . . . . 7.4.4 Schemes of higher-order approximation . . . . . . . . . . . . . . . . . .

211 212 213 214 215

8 Vector additive schemes

218

8.1 Vector schemes for first-order equations . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Vector differential problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Stability of vector additive schemes . . . . . . . . . . . . . . . . . . . . . 8.1.3 Stability with respect to the right-hand side . . . . . . . . . . . . . . .

218 218 220 223

8.2 Stability of vector additive schemes in Banach spaces . . . . . . . . . . . . . 8.2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Vector additive scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Study on stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

224 224 225 226

8.3 Schemes of second-order accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Three-level vector schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Schemes of the alternating triangle method . . . . . . . . . . . . . . .

228 228 229 231

8.4 Vector schemes for equations of second order . . . . . . . . . . . . . . . . . . . 8.4.1 The Cauchy problem for a second-order equation . . . . . . . . . . 8.4.2 Vector problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Scheme with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Stability of additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . .

232 232 234 235 236 238

xii 9

Contents

Iterative methods

240

9.1 Basics of iterative methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Simple iteration method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 The Chebyshev iterative method . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Two-level variation-type methods . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Conjugate gradient method . . . . . . . . . . . . . . . . . . . . . . . . . . . .

240 240 242 243 244 245

9.2 Iterative alternating direction method . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Iterative method with two-component splitting . . . . . . . . . . . . 9.2.2 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Modified iterative method of alternating directions . . . . . . . . . 9.2.4 Multicomponent splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

246 246 247 249 250

9.3 Iterative alternating triangle method . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Iterative method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 Convergence rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Modified iterative method of alternating triangles . . . . . . . . . .

252 252 253 255

9.4 Iterative cluster aggregation methods . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Transition to a system of equations . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Iterative method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3 Parallel variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Aggregation of unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 256 257 259 260

10 Splitting of the operator at the time derivative

263

10.1 Schemes with splitting of the operator at the time derivative . . . . . . . . 10.1.1 Preliminary discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.3 Vector problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.4 Vector additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

263 263 264 266 268 272

10.2 General splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Preliminary discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.3 Scheme with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4 Schemes with a diagonal operator . . . . . . . . . . . . . . . . . . . . . . . 10.2.5 The more general problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .

272 273 274 276 278 279

10.3 Explicit-implicit splitting schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Boundary value problems for systems of equations . . . . . . . . . 10.3.3 Schemes with a diagonal operator . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

282 282 283 285 289

Contents

11 Equations with pairwise adjoint operators

xiii 291

11.1 Splitting schemes for a system of equations . . . . . . . . . . . . . . . . . . . . . 11.1.1 Preliminary discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.3 A priori estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.4 Schemes with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.5 Splitting schemes to find the p-th component of the solution . 11.1.6 Additive schemes for systems of equations . . . . . . . . . . . . . . .

291 292 293 295 299 303 306

11.2 Additive schemes for a system of first-order equations . . . . . . . . . . . . . 11.2.1 Statement of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3 Schemes with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Explicit-implicit schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.5 Additive schemes of componentwise splitting . . . . . . . . . . . . . 11.2.6 Regularized additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . .

310 310 313 316 318 322 324

11.3 Another class of systems of first-order equations . . . . . . . . . . . . . . . . . 11.3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Scheme with weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Additive schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.4 More general problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.5 Problems of hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . .

326 326 328 330 333 335

Bibliography

339

Index

353

Notation

A, B, C , D, S

difference operators

E

identity operator



A

adjoint operator

A1

inverse of the operator A

A>0

positive operator (.Ay, y/ > 0, if y ¤ 0)

A0

non-negative operator (.Ay, y/  0)

A  ıE, ı > 0 1 A0 D .A C A / 2 1 A1 D .A  A / 2

positive definite operator

AD

p X

A˛

self-adjoint part of the operator A skew-symmetric part of the operator A p-componentwise splitting of the operator A

˛D1

H

finite-dimensional real Hilbert space

., /

scalar product in H

kk

norm in H

.y, v/A D .Ay, v/

scalar product in HA (operator A D A > 0)

k  kA

norm in HA

Lm .!/

Banach space of grid functions, m D 1, 2, 1

k  km

norm in Lm

kAk, kAkm

norm of a difference operator A

ŒA, m ŒA

logarithmic norm of a difference operator A

M , M˛

positive constants



computational domain

@

boundary 

r, grad

gradient

xvi

Notation

div, div v D r  v

divergence

!

set of interior grid points

@!

set of boundary points

h, h˛

steps of a spatial grid



time step

 , ˛

weights of a difference scheme

yx D

y.x C h/  y.x/ h

y.x/  y.x  h/ h 1 y ı D .yx C yxN / x 2 yx  yxN yxx N D h y D y n D y.x, t n /

yxN D

forward difference derivative at the point x backward difference derivative at the point x central difference derivative at the point x second difference derivative at the point x value of a grid function at the point x at the time level t n D n , n D 0, 1, : : :

Chapter 1

Introduction

Additive difference schemes result from a representation of an unsteady problem operator as the sum of operators with a simpler structure. The transition to a new time level is performed via solving a sequence of simpler problems. Such schemes are applied in different variants to solve numerically complex transient problems for PDEs. In this introductory part of the book, we present the key publications connected with the construction and investigation of additive difference schemes (schemes of splitting). The contents of the book are also briefly discussed.

1.1 Numerical methods To study up-to-date applied problems, information technologies that involve mathematical modeling as the key component are applied [8, 31, 45, 49, 97, 137, 171, 187]. Computational tools make it possible to describe properties of the object under consideration in detail and with the necessary completeness using adequate mathematical models that include systems of interconnected nonlinear time-dependent PDEs, ODEs and algebraic equations. The efficient solution of most applied problems requires the use of computers, and therefore we must develop numerical methods for computers [51, 70, 85, 92, 107]. In the numerical solving of unsteady problems, emphasis is on difference methods [6,36, 55, 63, 64, 76, 98, 110, 131, 166, 173, 181], where approximation in space may be both finite difference and finite element. The use of finite element procedures for solving transient problems is discussed in, e.g., [41, 182]. The theory of difference schemes [131] deals with issues of the construction and investigation of numerical algorithms for solving problems of mathematical physics. Theoretical studies in computational mathematics, which have been conducted by A. Samarskii and his followers [198] from the late 1940s up to the current moment, allow to develop a new direction in the theory of difference schemes that includes the formulation of the basic principles of difference schemes, mathematically rigorous proof of their stability and convergence as well as the corresponding algorithmic implementation. On the basis of these works, a great number of algorithms were created and applied for practical calculations to solve various problems of mathematical physics including thermo-physics, gas dynamics, magnetic gas dynamics, plasma physics, environment issues and other important scientific problems.

2

Chapter 1 Introduction

The theory of difference methods for solving problems of mathematical physics is developing in the following main directions: 

the construction of discrete analogs that inherit the basic properties of the original differential problem;



the study of stability (well-posedness) of difference problems;



an efficient computational implementation of the developed algorithms on modern computing systems.

To construct difference schemes [131, 132, 135], the general principle of conservatism has been formulated. A difference scheme is said to be conservative if it satisfies the corresponding conservation law at the discrete level. Homogeneous difference schemes have been obtained for problems with discontinuous coefficients as well as for problems with generalized solutions. To design a difference scheme on an arbitrary mesh, the method of support operators has been proposed. It is based on consistent approximations for differential operators of vector analysis (the gradient, divergence and curl). Samarskii has developed and applied the general methodological principle for obtaining difference schemes of prescribed quality that is based on small perturbations of operators (coefficients) of a difference scheme – the regularization principle for difference schemes. In the study of difference schemes for time-dependent problems of mathematical physics, the general theory of stability (well-posedness) for operator-difference schemes [131, 134, 136, 197] is in common use. At the present time, the exact (matching necessary and sufficient) conditions for stability are obtained for a wide class of two- and three-level difference schemes considered in finite-dimensional Hilbert spaces. We emphasize the constructive nature of the general theory of stability for operator-difference schemes, where stability criteria are formulated in the form of operator inequalities that are easy to verify. Among the most important generalizations, we highlight the use of the general theory of stability for ill-posed evolutionary problems [140, 141, 147, 155] and for investigating projection-difference schemes (finite element procedures) [142, 145, 197]. New a priori estimates for stability have been obtained in time-integral norms [156]. Using these, in particular, the convergence of difference schemes for problems with generalized solutions has been studied. Special attention should be given to a priori estimates for strong (coefficient) stability under various assumptions on the perturbation of operators (coefficients) of differential and difference problems [66, 157]. To find an approximate solution, we must solve large systems of linear or nonlinear algebraic equations. Iterative methods are widely applied to solve finite difference equations [114, 131, 138]. The problem of ordering of iteration parameters is successfully resolved for the Chebyshev iterative techniques. The optimization of choosing iteration parameters for numerical solving of non-self-adjoint problems has been performed at the general operator level. New versions of the iterative method of alternat-

Section 1.2 Additive operator-difference schemes

3

ing directions have been proposed. Special attention should be given to the alternating triangle iterative method, which belongs to the class of the fastest techniques and can be applied to general discrete elliptic equations. The constructive nature of general results, obtained in the theory of difference methods, results in the their wide use for solving large scientific and applied problems: predictions of nuclear and thermonuclear power plants, simulations of controlled thermonuclear fusion and so on. Difference methods are applied to the numerical study of heat and mass transfer phenomena, problems of continuum mechanics [139, 142]. In numerically solving initial-boundary value problems for multidimensional PDEs, great attention is paid to the construction of additive schemes. The transition to a chain of simpler problems allows us to construct efficient difference schemes – we speak of splitting with respect to the spatial variables. In some cases, it is useful to separate subproblems of distinct nature – we speak of splitting into physical processes. Such schemes appear in the solution of unsteady problems for systems of interconnected equations. In recent years, there has been an active discussion of regionally additive schemes (domain decomposition methods), which are focused on designing computational algorithms for parallel computers. In multicomponent splitting (into three or more operators), unconditionally stable additive schemes are obtained using the concept of summarized approximation, i.e., they are based on the transition to a chain of individual problems associated with particular operator terms. In some cases, multicomponent additive schemes are constructed without employing the concept of summarized approximation. All the issues above are reflected in the present book with various degrees of detail.

1.2 Additive operator-difference schemes This work deals with some approaches to the construction of numerical methods for solving the Cauchy problem for evolutionary equations of first and second order considered in a finite-dimensional Hilbert space H . An example is the problem du C Au D f .t /, t > 0, (1.1) dt (1.2) u.0/ D u0 , where f .t /, u0 are given, and u.t / is a searched function with values in the finitedimensional Hilbert space H . Let the operator A be constant (independent of t ) and non-negative in H (A  0). Using the form (1.1), (1.2), we can formulate the Cauchy problem for a system of linear first-order ODEs m dui .t / X aij .t /uj .t / D fi .t /, t > 0, C dt ui .0/ D

j D1 u0i ,

i D 1, 2, : : : , m.

4

Chapter 1 Introduction

Here u D ¹u1 , u2 , : : : , um º is the vector of unknowns, f D ¹f1 , f2 , : : : , fm º stands for the vector of the prescribed right-hand sides, and A D ¹aij º is a matrix with elements aij , i , j D 1, 2, : : : , m. The formulation (1.1), (1.2) appears after discretization in space during numerical solving of initial-boundary value problems for equations of mathematical physics. In finite-difference methods [131], u is a grid function defined at the nodes of a computational grid. For finite element procedures [102, 172], unknowns u are coefficients of the decomposition of the approximate solution over a finite elemental basis. To solve numerically the problem (1.1), (1.2), we use, for simplicity, a uniform grid with a step  > 0 and let t n D n , n D 0, 1, : : : , y n D y.t n /. The explicit scheme is the most simple for computational implementation: y nC1  y n C Ay n D f n , n D 0, 1, : : : .  However, explicit schemes are conditionally stable, and their stability is valid under some constraints on the time step. In the case of the above explicit scheme, these restrictions have the form 2 .  kAk Here we focus on unconditionally stable difference schemes. The fully implicit scheme provides an example: y nC1  y n (1.3) C Ay nC1 D f nC1 , n D 0, 1, : : : .  The computational implementation of implicit schemes is much more complicated in comparison with explicit schemes. To determine the approximate solutions at the new time level in the case (1.3), we solve the problem .E C A/y nC1 D y n C f nC1 . The inversion of the operator E C A may be very difficult. Thus, it seems natural to construct difference schemes for unsteady problems such that they will be unconditionally stable, but, at the same time, their implementation would be considerably simpler than in the case of implicit schemes. The most interesting results have been obtained taking into account a special structure of the problem operator A. We define a class of the problems (1.1), (1.2), where the operator A has the following p-component additive representation: AD

p X

A˛ .

(1.4)

˛D1

Assume that the operators A˛ , ˛ D 1, 2, : : : , p are simpler than A. We organize computations in such a way that the transition to a new time level in solving the problem

5

Section 1.2 Additive operator-difference schemes

(1.1), (1.2) is not more complicated than the solution of the p problems for the individual operator terms: A˛ , ˛ D 1, 2, : : : , p, i.e., for the equations of type du˛ C A˛ u˛ D f˛ .t /, dt

t > 0,

˛ D 1, 2, : : : , p.

Such difference schemes are called additive difference schemes. The simplest example of additive schemes with two-component splitting A D A1 C A2

(1.5)

is the explicit-implicit scheme y nC1  y n C A1 y nC1 C A2 y n D f n , 

n D 0, 1, : : : ,

where only a part of the problem operator A is shifted to the upper time level. An additive difference scheme is defined, on the one hand, by the choice of the splitting (1.4), i.e., by specifying individual operator terms A˛ , ˛ D 1, 2, : : : , p, and on the other hand, it depends on the construction of supplementary problems, i.e., on the arrangement of calculations via solving simpler subproblems. Let us highlight some general possibilities to select the operators A˛ , ˛ D 1, 2, : : : , p. In this book, we will discuss this issue in detail considering comprehensive examples. Economical difference schemes provide the classical examples of additive difference schemes applied to multidimensional problems of mathematical physics. First of all, we speak of the well-known ADI methods [28, 104], where an operator of the two-dimensional problem is represented as the sum of two one-dimensional terms. In this case, we have the additive difference schemes of splitting with respect to spatial directions. In many cases, the operator terms are attributed to the description of the processes with distinct nature. For example, in problems of continuum mechanics, we predict transport processes of a medium where we can separate the convective transport of the medium (the operator A1 ) from the diffusive transport (the operator A2 ). To emphasize this feature of the problem, we speak of splitting into physical processes. In designing numerical algorithms for solving time-dependent problems with PDEs on modern parallel computers, we are oriented to the use of domain decomposition (splitting) methods [96,108,184]. An individual computational node (processor) solves a boundary value problem in a separate subdomain. In this case, we have splitting with respect to subdomains. The corresponding additive difference schemes are called regionally additive. We emphasize the following aspects of our study. The investigation of additive schemes for problems of type (1.1), (1.2), (1.4) is conducted using the general theory of stability for operator-difference schemes [131, 134, 136] under some weak restrictions on the operator terms A˛ , ˛ D 1, 2, : : : , p. For instance, the main results may be obtained for non-negative and, in general, pairwise noncommutative operators A˛ ,

6

Chapter 1 Introduction

˛ D 1, 2, : : : , p. Particular peculiarities of the operators are taken into account in selecting estimates for convergence of additive difference schemes. In this work, we formulate general conditions for stability and convergence of various classes of additive schemes, whereas illustrative examples of specific constructing additive schemes are presented separately.

1.3 The main results of the theory of additive operator-difference schemes The history of additive difference schemes began with the works [28,104], where twodimensional parabolic problems were solved using the ADI methods. These schemes belong to the class of economical difference schemes. They employ the transition to a new time level, where the number of arithmetic operations on one node of a computational grid does not depend on the total number of grid points, similarly to the simplest explicit difference schemes. In the alternating direction method, the terms A˛ , ˛ D 1, 2 in the representation (1.5) are one-dimensional difference operators. In the Douglas–Rachford scheme, the solution at a new time level is evaluated from the equations y nC1=2  y n C A1 y nC1=2 C A2 y n D ' n , 

(1.6)

y nC1  y n (1.7) C A1 y nC1=2 C A2 y nC1 D ' n ,  where ' n D f .t nC1=2 /. This scheme is unconditionally stable (for any  > 0) for nonnegative operators A˛ , ˛ D 1, 2, and the difference solution satisfies the levelwise estimate   .E C A2 /y nC1   k.E C A2 /y n k C  k' n k. The ADI methods are widely used in computational practice. Their application is essentially based on the two-component decomposition (1.5). Many works are devoted to studying new variants of the ADI schemes and techniques that are close to them. In our investigation, we omit any review of relevant works referring the reader to the books [30, 93, 128, 130] and to the references in them. The fundamental problem is the construction of alternating direction schemes that are based on the general multicomponent splitting (1.4). For example, in constructing economical difference schemes for solving three-dimensional problems, we have splitting into three terms: A D A1 C A2 C A3 . It is easy to write a splitting scheme which is a direct generalization of the scheme (1.6), (1.7) y nC1=3  y n C A1 y nC1=3 C A2 y n C A3 y n D ' n , 

7

Section 1.3 The main results

y nC2=3  y n C A1 y nC1=3 C A2 y nC2=3 C A3 y n D ' n ,  y nC1  y n C A1 y nC2=3 C A2 y nC2=3 C A3 y nC1 D ' n .  However, unconditional stability can be proved only under the additional and very strong restrictions of the commutativeness of the operator terms A˛ , ˛ D 1, 2, 3. To construct economical difference schemes for the problem (1.1)–(1.3), N. Yanenko has proposed the method of fractional steps [215, 218]. In the fully implicit scheme of componentwise splitting, the approximate solution is determined from solving one-dimensional problems y nC˛=p  y nC.˛1/=p C A˛ y nC˛=p D '˛n , ˛ D 1, 2, ..., p  using the splitting of the right-hand side: p X 'n D '˛n , n D 0, 1, : : : .

(1.8)

(1.9)

˛D1

The stability of the scheme (1.8), (1.9) is checked directly. However, there remains the problem of convergence of the difference solution to the exact one due to the fact that the difference equations (1.8) do not approximate the original equation (1.1). The simplest approach is connected with the elimination of the intermediate grid functions y nC˛=p , ˛ D 1, 2, : : : , p  1 and the following consideration of the scheme that involves integer time steps. But this way does not always lead to the necessary goal, and it almost always becomes complicated and difficult to implement. The fundamental improvement was made by Samarskii in the work [116], where, for studying additive difference schemes, the concept of summarized approximation was introduced. The chain of equations (1.8) approximates equation (1.1) in a summarized sense. Namely, let ˛n D O.1/ be the truncation error of the corresponding difference equation. The difference scheme (1.8), (1.9) approximates equation (1.1) if n D P 2 n ˛D1 ˛ ! 0 as the computational grid becomes more fine. Using the concept of summarized approximation, we can prove the convergence of componentwise splitting schemes in Hilbert and Banach spaces of grid functions by means of deriving estimates for stability with respect to the initial data and the right-hand side for the individual equations of the scheme (1.8), (1.9). To construct schemes with second-order accuracy with respect to  (see [38, 39, 170]), the idea of symmetrization is applied. The improvement in accuracy is achieved via replacement of the sequence of problems A1 7! A2 7!    7! Ap by the chain A1 7! A2 7!    7! Ap 7! Ap 7! Ap1 7!    7! A1 during the transition to a new time level.

8

Chapter 1 Introduction

Based on the idea of summarized approximation, locally one-dimensional difference schemes for numerical solving of boundary value problems for parabolic equations and systems were constructed (see [117,119,120,122]). Special attention should be given to the development of additive difference schemes for hyperbolic equations [71, 72, 121, 123]. Among many studies on the construction of various schemes of componentwise splitting, we emphasize the paper [50], where the approximate solution of the problem (1.1), (1.2), (1.4) is determined from y˛nC1  y n C A˛ y˛nC1 D '˛n , ˛ D 1, 2, ..., p, p p 1 X nC1 y , n D 0, 1, : : : . y nC1 D p ˛D1 ˛ The basic potential advantage of this additively averaged difference scheme results from the fact that the auxiliary quantities y˛nC1 , ˛ D 1, 2, : : : , p can be calculated independently. This is especially important for the construction of computational algorithms oriented to modern parallel computers. The next important step in the theory of additive difference schemes has been made in the works [1, 2]. Instead of finding the scalar function u from (1.1), (1.2), (1.4), we search the vector u D ¹u1 , u2 , : : : , up º. Each individual component is defined as the solution of the similar problems p X du˛ Aˇ uˇ D f .t /, C dt ˇ D1 0

u˛ .0/ D u ,

t > 0,

˛ D 1, 2, : : : , p.

To solve this system of equations, it is possible to employ, e.g., the following vector additive difference scheme: p ˛ X X y˛nC1  y˛n Aˇ yˇnC1 C Aˇ yˇn D ' n , C  ˇ D1

n D 0, 1, : : : ,

ˇ D˛C1

˛ D 1, 2, : : : , p.

In this case, we have a full approximation scheme, where each difference equation approximates equation (1.1). Vector additive schemes allow to construct easily the schemes of higher accuracy. In addition, without essential complications, we can consider additive schemes for evolutionary equations of second order, i.e., for the Cauchy problem for the equation d 2u C Au D f .t /, dt 2 with the splitting (1.4).

t >0

Section 1.3 The main results

9

The regularization principle for difference schemes [126] is the general methodological principle for obtaining difference schemes with prescribed quality. It employs the general results of the stability theory for operator-difference schemes and involves perturbations of the coefficients of the difference scheme. On its basis, stable difference schemes for many problems of mathematical physics were obtained, new classes of monotone difference schemes were constructed, and efficient iterative methods were investigated. This general approach is used to obtain additive difference schemes. Employing the regularization principle, a new class of additive schemes with full approximation has been developed [150]. For (1.1), (1.2), we write the scheme y nC1  y n C .E C A/1 Ay n D f n , n D 0, 1, : : : ,  which is slightly different from the fully implicit scheme (1.3) in the right-hand side. In this case, unconditional stability is ensured by a multiplicative perturbation of the operator A. In a similar way, for the problem (1.1), (1.2), (1.4), we can construct the regularized additive difference scheme 2 y nC1  y n X .E C  A˛ /1 A˛ y n D f n , n D 0, 1, : : : , C  ˛D1 which is stable under the restriction   0.5p. This scheme is based on a multiplicative perturbation of each operator term in the decomposition (1.4). In this way, we can obtain regularized additive difference schemes of second-order accuracy with respect to  for evolutionary equations of second order. The issues of the construction, investigation and application of additive difference schemes are reflected in many publications. Here emphasis is only on the problems, which, on the one hand, are, in our view, of a fundamental nature, and on the other hand, meet the content of this book in the best way. The elementary level in the theory and practice of splitting schemes is presented in some textbooks on numerical analysis of unsteady problems for PDEs. The following books should be mentioned among others: [6,181]. A more comprehensive discussion is available in [63, 131]. Special attention should be given to the works reflecting the advanced level of investigations. The first book from the available list is [218]. The book was published in Russian in 1967. More recent researches on splitting schemes are presented in [94] (Russian edition [93]). The most complete presentation of the theory of additive schemes is reported in the book [151]. In this book, in particular, new classes of splitting schemes (vector additive schemes, regularized additive schemes) are described. Unfortunately, this book was published in Russian and so it is almost unknown to the international scientific community. Below there are several recently published books concerned with splitting schemes. The splitting technique for solving multiphysics problems is discussed in [44]. The

10

Chapter 1 Introduction

author describes the splitting methods used to decouple equations which can be analyzed separately. A new iterative operator splitting method is proposed to improve the quality of the approximate solution. Iterative splitting schemes can be extended to different applications. The material in the book [34] is connected with studying additive schemes of componentwise splitting for the evolutionary equation of first order. The results of solving the system of equations for convection-diffusion-reaction (an airpollution model) are presented. The authors of [61] consider the splitting schemes for solving convection-diffusion problems. The main results are concerned with studying the convergence of the approximate solution to the exact one for nonlinear equations and systems.

1.4 Contents of the book The theory of additive difference schemes is based on the general theory of stability (well-posedness) for operator-difference schemes. The main results of this theory are summarized in Chapter 2. The Cauchy problems for operator-differential equations of first and second order are considered in a finite-dimensional Hilbert space. We present the basics of the theory of operators in such spaces, which serve as the main mathematical tool in the stability theory for the operator-difference schemes. To solve numerically unsteady problems, two- and three-level difference schemes are constructed, which are supplemented with stability conditions that are formulated in the form of easy verifying operator inequalities. Special attention should be given to stability criteria for the schemes with weights. A brief discussion is presented for studying difference schemes in Banach spaces of grid functions. Chapter 3 deals with some examples of how to select the splitting of the operator A into the individual operator terms A˛ , ˛ D 1, 2, : : : , p. Additive difference schemes are used to solve numerically multidimensional transient problems of mathematical physics, where one-dimensional problems are the most simple ones. On the basis of splitting with respect to the spatial variables, we construct locally one-dimensional schemes for a parabolic equation, transport equation, hyperbolic second-order equation and their systems. In using computational algorithms of domain decomposition, which focus on modern parallel computers, the original problem is divided into several subproblems, each of which is solved in its own subdomain on its individual processor. Various versions of operators of domain decomposition are proposed to construct regionally additive difference schemes for solving time-dependent problems of mathematical physics. Examples are also considered where individual operator terms have a distinct nature – we speak of splitting into physical processes. In solving an unsteady problem of convection-diffusion, additive difference schemes are based on separation of the diffusion process from the convection phenomenon. Special attention is given to solving systems of equations. In this case, splitting makes it possible to solve a se-

Section 1.4 Contents of the book

11

quence of problems formulated for the individual problem, weakly coupled with each of the other problems. Subsequent chapters discuss the basic classes of additive operator-difference schemes for evolutionary equations of first and second order. The study is conducted in sufficiently general conditions involving weak constraints on operators of the problem under consideration. We formulate the most important results with minimal mathematical tools. Specific features and common characteristics of different classes of additive operator-difference schemes are investigated. Chapter 4 is devoted to the most well-studied class of two-component additive schemes, where an operator of the problem is split into the sum of two operators. The classical ADI methods belong to this class of additive schemes. These schemes are closely connected with factorized additive difference schemes. A detailed investigation is performed for a special class of two-component additive splitting schemes for problems with a self-adjoint operator – we speak of additive schemes of the alternating triangle method. Their consideration is based on the direct use of the general theory of stability for operator-difference schemes. Additive difference schemes are designed for evolutionary equations of second order along with factorized schemes of multicomponent splitting. Unconditionally stable additive schemes of multicomponent splitting for arbitrary noncommutative operators are based on the concept of summarized approximation. Chapter 5 investigates various approaches to numerically solving the Cauchy problem for an evolutionary equation on the basis of the solution of intermediate problems for the individual operator terms. A comparative study of the standard schemes of componentwise splitting and additively averaged schemes of componentwise splitting is conducted. Estimates for stability in finite-dimensional Hilbert and Banach spaces are derived using the appropriate a priori estimates for the intermediate problems. The construction and study of additive schemes of componentwise splitting with secondorder accuracy is briefly discussed. Chapter 6 demonstrates some results on the construction of stable difference schemes based on the regularization principle for difference schemes, i.e., via perturbations of the difference scheme operator. Additive and multiplicative regularized difference schemes are constructed for the evolutionary first-order equation. Constructive possibilities of the regularization principle for difference schemes are shown in designing stable additive difference schemes with multicomponent splitting. Regularized additive schemes of second-order accuracy are constructed, too. Regularized additive difference schemes for second-order equations are discussed. Investigations on stability and convergence of additive operator-difference schemes for transient problems may be conducted (see Chapter 7) on the basis of an operator of the transition from the current time level to the next one. The classical scheme of multicomponent splitting corresponds to the use of a multiplicative transition operator. A new class of splitting schemes that are based on an additive representation of the transition operator is introduced. Some possibilities of constructing splitting schemes

12

Chapter 1 Introduction

of the second order in time are analyzed. The construction of inhomogeneous operatorsplitting schemes is also discussed in which different types of transition operators are used for the individual splitting operators. In Chapter 8, we study vector additive difference schemes of multicomponent splitting. The original problem is reformulated as a vector problem. In this case, instead of a single approximate solution, we search for a vector of approximate solutions. The corresponding additive schemes are schemes of full approximation – at each time step we search for the approximate solution of the problem. The construction of vector additive difference schemes is conducted on a single methodological basis using results of the general theory of stability for operator-difference schemes and the regularization principle. Two- and three-level vector additive schemes are applied to evolutionary equations of first and second order. Iterative methods for solving steady-state problems of mathematical physics are often treated as pseudo-time evolution methods for solving time-dependent problems. Many iterative methods may be associated with the use of certain additive schemes. Chapter 9 presents the basic facts of the theory of iterative methods and discusses the possibility of constructing iterative methods via splitting the problem operator into the sum of simpler operators. Factorized methods are briefly discussed including the iterative method of alternating directions. Nowadays, much attention should be given to the class of iterative alternating triangle methods. On the basis of additive difference schemes, we develop new classes of general iterative methods that are associated with a grouping of equations or unknowns. Block iterative methods and the alternating Schwarz method in different variants provide examples of these techniques. Classical additive operator-difference schemes for evolutionary equations are constructed by means of additive splitting of the leading operator (associated with the unknown variable) into several terms. For a number of applied problems, the operator at the time derivative has an additive representation. In Chapter 10, we investigate this new class of evolutionary problems and design vector additive operator-difference schemes. Some generalization of the results are discussed, too. The main theoretical results on stability and convergence of additive operator-difference schemes are obtained for scalar evolutionary first-order equations, and, in some cases, for second-order equations. For computational practice, considerable interest is connected with operator-splitting schemes for systems of evolutionary equations. For example, in vector problems, the individual components of the vector of unknowns are interconnected with each other. In this case, the use of certain splitting schemes is intended to get a simple problem for the individual components of the solution at a new time level. Chapter 11 deals with the Cauchy problem for a special linear system of first-order equations with conjugate operators in a Hilbert space. Such a structure is typical for equations, in particular, describing problems of acoustics (dynamics of a compressible fluid) and electrodynamics. Some other systems of equations are also considered that are typical, e.g., for problems of incompressible fluid dynamics and porous media flows with consolidation.

Section 1.4 Contents of the book

13

The main conclusion of our study consists in the statement that additive operatordifference schemes (operator-splitting schemes) represent a very powerful and, in some cases, practically indispensable mathematical tool for constructing efficient numerical algorithms for numerically solving various problems for PDEs and their systems.

Chapter 2

Stability of operator-difference schemes

We consider the Cauchy problems for operator-differential equations of the first and second order in time. To solve them numerically, two- and three-level difference schemes are constructed. The present study is based on the general stability theory for operator-difference schemes in Hilbert spaces of grid functions. The emphasis is on deriving criteria for stability of schemes with weights. Peculiarities of difference schemes under examination in Banach spaces of grid functions are highlighted.

2.1 The Cauchy problem for an operator-differential equation Difference schemes for problems of mathematical physics, as a rule, are investigated in Hilbert spaces of grid functions [131]. Therefore, below we provide the basics of functional analysis, which are used in the theory of difference schemes to treat operators in finite-dimensional Hilbert spaces. First, we consider the Cauchy problem for an operator-differential equation of the first order in time. Next, elementary a priori estimates will be derived for its solution. They will serve us as a reference point in constructing operator-difference schemes.

2.1.1 Hilbert spaces Consider a real linear space [87,90] H equipped with operations of addition and scalar multiplication by a real number, i.e., for each pair of y 2 H , v 2 H , we assign the element y Cv 2 H , and, for every y 2 H and any real , we have y 2 H . Moreover, for elements y, v, z of H and real numbers ,  the following axioms hold: (1) y C v D v C y, y C .v C z/ D .y C v/ C z; (2) .y/ D ./y; (3) .y C v/ D y C v, . C /y D y C y; (4) there exists a null element 0 such that y C 0 D y for every y 2 H ; (5) for every y 2 H there exists a unique element .y/ 2 H such that yC.y/ D 0; (6) 1  y D y.

Section 2.1 The Cauchy problem for an operator-differential equation

15

Elements yi , i D 1, 2, : : : , m of a linear space H are called linearly independent, if from the equality m X i yi D 0 iD1

it follows that i D 0, i D 1, 2, : : : , m. Otherwise the elements yi , i D 1, 2, : : : , m are called linearly dependent. A space H is said to be k-dimensional if in H there exist k linearly independent elements and any .k C 1/-st element is linearly dependent. if A nonempty closed space H1 of elements of a linear space H is called a subspace P  y and only if yi , i D 1, 2, : : : , m 2 H1 implies that every linear combination m i i iD1 is also in H1 . A linear space H is called a normed linear space if for any element y 2 H , there is defined a real number kyk, called a norm, that satisfies the conditions: (1) kyk  0, and kyk D 0 if and only if y D 0; (2) kyk D jj kyk; (3) ky C vk < kyk C kvk (the triangle inequality). A linear space may be normed in a different way. Let kyk1 and kyk2 be norms in H . If there exist constants 0 < M1  M2 such that M1 kyk1  kyk2  M2 kyk1 for all y 2 H , then these norms are called equivalent. In a finite-dimensional space any two norms are equivalent. A sequence yi of elements of a linear normed space H is said to converge to an element y 2 H if kyi  yk ! 0 as i ! 1. If kyi  yj k ! 0 as i , j ! 1, then the sequence yi is called a Cauchy sequence. A linear normed space H is referred to as complete if every Cauchy sequence yi from this space converges to an element y of H . A complete linear normed space is called a Banach space. Any finite-dimensional linear normed space is complete. Consider a real linear space H and assume that for any elements y, v of H , a scalar product .y, v/ is defined such that (1) .y, v/ D .v, y/; (2) .y C v, z/ D .y, z/ C .v, z/; (3) .y, v/ D .y, v/; (4) .y, y/  0, and .y, y/ D 0 if and only if y D 0. A real linear normed space H with the norm induced by the scalar product kyk D .y, y/1=2 is called a unitary space. A complete unitary space is said to be a Hilbert space. Any finite-dimensional unitary space is complete.

16

Chapter 2 Stability of operator-difference schemes

Two elements y, v of a unitary space are called orthogonal if .y, v/ D 0. A system of elements yi , i D 1, 2, : : : , m of H is called an orthonormal system if .yi , yj / D ıij , i , j D D 1, 2, : : : , m, where ² 1, i D j , ıij D 0, i ¤ j is Kronecker’s symbol. Elements y, v of a unitary space satisfy the Cauchy–Schwarz inequality, i.e., j.y, v/j  kyk kvk, with equality if and only if y and v are linearly dependent. We also recall the parallelogram identity ky C vk2 C ky  vk2 D 2.kyk2 C kvk2 /, and the identity

1 .y, v/ D .ky C vk2  ky  vk2 /, 4 where y, v are elements of a unitary space.

2.1.2 Linear operators in a finite-dimensional space Throughout the following we assume that H is a finite-dimensional linear normed space. We say that A is an operator acting from a set D  H onto some other set R  H if for every element y from D there exists a corresponding element v from R. The set D is called the domain of the operator A (is denoted by D.A/). The set of elements v D Ay 2 R is referred to as the range of the operator A (R.A/). We denote the null operator by 0 and the identity operator by E. We say that an operator A acts in a space H if D.A/ D H and R.A/  H . Let an operator A map D.A/ onto R.A/ in a one-to-one manner. Then it is possible to define the operator A1 acting from R.A/ onto D.A/ such that A1 v D y if and only if Ay D v. The operator A is called nondegenerate and A1 is the inverse of A. An operator A is said to be linear if for all y, v 2 D.A/ and real numbers ,  A.y C v/ D Ay C Av. For a linear operator A in a finite-dimensional linear space H , there exists the inverse of A with the domain D.A1 / D H if and only if equation Ay D 0 has a unique solution y D 0. A linear operator A acting in H is called bounded if there exists a constant M > 0 such that kAyk  M kyk for all y 2 H . In a finite-dimensional space, any linear operator is bounded.

Section 2.1 The Cauchy problem for an operator-differential equation

17

The smallest constant M from the above inequality is called a norm of the operator A and is denoted by kAk. From this definition, it follows that kAyk D sup kAyk. y¤0 kyk kykD1

kAk D sup

The norm satisfies the following properties: (1) kAk D jj kAk; (2) kA C Bk  kAk C kBk; (3) kABk  kAk kBk . If .AB/y D .BA/y for all y 2 H , then the operators A and B are called commutative .AB D BA/. The bounded operator A1 exists if and only if there exists a constant ı > 0 such that kAyk  ıkyk for all y 2 H and kA1 k  ı 1 .

2.1.3 Operators in a finite-dimensional Hilbert space An operator A is called the adjoint operator to A in a Hilbert space H if, for all y, v 2 H , the following identity holds: .Ay, v/ D .y, A v/. For a bounded linear operator A with the domain D.A/ D H , there exists a unique operator A with the domain D.A / D H . The following equalities are satisfied: .A / D A,

.A C B/ D A C B  ,

.AB/ D B  A .

The operator A is linear and bounded, and kA k D kAk, moreover, kAk D kA Ak1=2 . An operator A acting in H is called self-adjoint if A D A. Let A and B be selfadjoint operators, then AB is self-adjoint if and only if the operators A and B are commutative. An operator A is called skew-symmetric if and only if A D A. Any operator A may be represented as the sum of a self-adjoint operator and a skew-symmetric operator: A D A0 C A1 , where

1 1 A0 D .A C A /, A1 D .A  A /. 2 2 An operator A in a finite-dimensional real Hilbert space H is called:

18

Chapter 2 Stability of operator-difference schemes



non-negative (A  0) if .Ay, y/  0 for all y 2 H ;



positive (A > 0) if .Ay, y/ > 0 for all y 2 H , except y D 0;



positive definite (A  ıE) if there exists a ı > 0 such that for all y 2 H , we have .Ay, y/  ıkyk2 .

By definition, the inequality A  B means that A  B  0. If an operator A acting in a finite-dimensional Hilbert space H is positive, then there exists the operator A1 . For a positive definite operator A, we have kA1 k  ı 1 . The product of two commutative, non-negative and self-adjoint operators A and B is also a non-negative self-adjoint operator. For any linear A, operators A A and AA are non-negative, and they are positive for positive A. An operator B is called the square root of an operator A if and only if B 2 D A .B D A1=2 /. For any non-negative self-adjoint operator A, there exists a unique non-negative self-adjoint square root that commutes with every operator commutative with A. For any self-adjoint non-negative operator A, we have the generalized Cauchy– Schwarz inequality .Ay, v/2  .Ay, y/.Av, v/. Let D be a self-adjoint positive (non-negative) operator in H . Then we can introduce the energy space HD consisting of elements H with the scalar product .y, v/D D .Dy, v/ and the norm (seminorm) kykD D .Dy, y/1=2 . For any A D A > 0, we have the inequality j.y, v/j  kykA kvkA1 . For a non-negative operator A, the number .Ay, y/ is called the energy of the operator. If there exist constants 2  1 > 0 such that for linear operators A and B the inequalities 1 B  A  2 B hold, then such operators are said to be energy equivalent. If ıE  A  E, then the numbers ı and are called the bounds of the operator A.

19

Section 2.1 The Cauchy problem for an operator-differential equation

2.1.4 The Cauchy problem for an evolutionary equation of first order Consider the Cauchy problem for an operator-differential equation of first order du C Au D f .t /, dt u.0/ D u0 ,

t > 0,

(2.1) (2.2)

where f .t / and u0 are given, and u.t / is a function with values in a finite-dimensional Hilbert space H . Let us obtain an elementary a priori estimate for the solution of the problem (2.1), (2.2), which expresses stability with respect to the initial data and the right-hand side. Theorem 2.1. Assume that A  0, then for the solution of the problem .2.1/, .2.2/, the following estimate holds: Z t 0 kf . /kd . (2.3) ku.t /k  ku k C 0

Proof. Multiplying equation (2.1) by u.t /, we get the equality   du , u C .Au, u/ D .f , u/. dt We have

 d 1d du ,u D kuk2 D kuk kuk, dt 2 dt dt .f , u/  kf k kuk, 

and, in view of the non-negativity of the operator A, we go to the inequality d kuk  kf k. dt Therefore, the estimate (2.3) follows immediately. Remark. Under the conditions of Theorem 2.1, it is often more convenient to consider an a priori estimate for the squared norm of the solution. Taking into account 1 1 .f , u/  kf k2 C kuk2 , 2 2 we obtain the inequality d kuk2  kuk2 C kf k2 dt and so   Z t 2 0 2 2 exp. /kf . /k d . (2.4) ku.t /k  exp.t / ku k C 0

The proof of (2.4) is based on the following simple version of Gronwall’s lemma.

20

Chapter 2 Stability of operator-difference schemes

Lemma 2.1. Assume that a function g.t / satisfies the inequality dg  ag.t / C b.t / t > 0 dt with a D const , b.t /  0, then the following estimate is valid:   Z t exp.a /b. /d . g.t /  exp.at / g.0/ C 0

Remark. Under the more general conditions A  ıE, ı D const , in view of Gronwall’s lemma, the solution of (2.1), (2.2) meets the a priori estimate   Z t 0 exp.ı /kf . /kd . ku.t /k  exp.ıt / ku k C 0

The above estimates for stability with respect to the initial data, which were obtained for the solution of (2.1), (2.2), will serve us as a guide in constructing and investigating operator-difference schemes arising after discretization in time.

2.1.5 Systems of linear ordinary differential equations Consider the Cauchy problem for a system of linear ordinary differential equations (ODEs) of first order as a typical example of the problem (2.1), (2.2). Let u D ¹u1 , u2 , : : : , um º be a vector of unknowns and f D ¹f1 , f2 , : : : , fm º be a specified vector of the right-hand sides for the equation dui .t / X aij .t /uj .t / D fi .t /, C dt m

ui .0/ D

j D1 u0i ,

i D 1, 2, : : : , m.

t > 0,

(2.5) (2.6)

Using vector notation, it is possible to rewrite the problem (2.5), (2.6) as the Cauchy problem for the following single equation: du C Au D f .t /, dt u.0/ D u0 ,

t > 0,

(2.7) (2.8)

where A D ¹aij º is a matrix with elements aij , i , j D 1, 2, : : : , m. The matrix A may be treated as a linear operator Pm in the finite-dimensional Hilbert (Euclidean) space H D l2 , where .y, v/ D iD1 yi vi is the scalar product and kyk D .y, y/1=2 is the norm.

Section 2.1 The Cauchy problem for an operator-differential equation

21

2.1.6 A boundary value problem for a one-dimensional parabolic equation Operator-differential equations appear in time-dependent problems of mathematical physics as the result of discretization in space (finite difference or finite element). Here we present an illustrative example of a boundary value problem for a one-dimensional parabolic equation. More informative multidimensional problems, which are the object of our investigation, will be considered later. Let u.x, t / be the solution of the equation   @ @u @u  k.x/ D f .x, t /, 0 < x < l, t > 0, (2.9) @t @x @x supplemented with the trivial boundary and initial conditions: u.0, t / D 0,

u.l, t / D 0,

u.x, 0/ D u .x/, 0

t > 0,

(2.10)

0 < x < l.

(2.11)

N D Œ0, l: Denote by !N the uniform grid with a step h on the interval  !N D ¹x j x D xi D ih,

i D 0, 1, : : : , N ,

N h D lº,

where ! and @! are the sets of interior and boundary points, respectively. The expansion of u in a Taylor series about any interior point x D xi yields ui˙1 D ui ˙ h

du h2 d 2 u h3 d 3 u .x / ˙ .xi / C O.h4 / .xi / C i dx 2 dx 2 6 dx 3

for a sufficiently smooth function u.x/. We use here the notation ui D u.xi /. For the so-called backward difference derivative (omitting the index i , i.e., using the indexfree notation of the theory of difference schemes), we have ui  ui1 h d 2u du .xi / C O.h2 /. D .xi /  h dx 2 dx 2 Thus, the backward difference derivative uxN approximates the first-order partial derivative du=dx with the first order (with the local truncation error O.h/ at each interior grid point) if u.x/ 2 C .2/ ./. Similarly, for the forward difference derivative, we obtain uiC1  ui h d 2u du .xi / C O.h2 /. ux  D .xi / C h dx 2 dx 2 Using the three-point stencil (with the points xi1 , xi , xiC1 ), we can introduce the central difference derivative uxN 

uı  x

uiC1  ui1 h2 d 3 u du .xi / C O.h3 /, D .xi / C 2h dx 6 dx 3

which approximates du=dx with the second order if u.x/ 2 C .3/ ./.

22

Chapter 2 Stability of operator-difference schemes

For the second-order derivative d 2 u=dx 2 , similar calculations give us the formula ux  uxN uiC1  2ui C ui1 . D h h2 This difference operator approximates the second-order partial derivative at the grid point x D xi with the second order if u.x/ 2 C .4/ ./. On the set of grid functions that equal zero on the set of boundary points (see (2.10)), we define the difference operator uxx N D

x 2 !.

Ay D .ayxN /x ,

(2.12)

If the coefficient k.x/ is sufficiently smooth, then it is possible to employ the formulas: ai D ki1=2 D k.xi  0.5h/, ki1 C ki ai D ,  2  1 1 1 C . ai D 2 ki1 ki The problem (2.9)–(2.11) is associated with the Cauchy problem for the operatordifferential equation dy C Ay D f , x 2 !, t > 0, dt y.x, 0/ D u0 .x/, x 2 !.

(2.13) (2.14)

We define the Hilbert space H D L2 .!/ on the set of grid functions y.x/ D 0, x … ! with the scalar product X .y, v/ D y.x/v.x/h. x2!

The difference operator A is self-adjoint in H (A D A ). The equality .Ay, v/ D .y, Av/ is checked immediately. In addition (see [131] for more details), with the standard constraint k.x/  , the operator A has a lower bound A  0 E, where 0 is the minimal eigenvalue of the difference operator of the second-order derivative: 4 8

h > 2. 0 D 2 sin2 h 2l l Therefore, we have for the solution of the problem (2.13), (2.14) the estimate   Z t exp. 0  /kf .x,  /kd . ky.x, t /k  exp. 0 t / ku0 .x/k C 0

Section 2.1 The Cauchy problem for an operator-differential equation

23

2.1.7 Equations of second order In addition to operator-differential equations of the first order in time, considerable attention should be given to second-order equations. Let us consider the Cauchy problem d 2u C Au D f .t /, t > 0, dt 2 du .0/ D v 0 . u.0/ D u0 , dt

(2.15) (2.16)

The problem (2.15), (2.16), e.g., results from discretization in space of a boundary value problem for the one-dimensional wave equation   @ @u @2 u  k.x/ D f .x, t /, 0 < x < l, t > 0, @t 2 @x @x u.0, t / D 0, u.l, t / D 0, t > 0, @u u.x, 0/ D u0 .x/, .x, 0/ D v 0 .x/, 0 < x < l. @t Let us obtain an elementary a priori estimate for the solution of the problem (2.15), (2.16), assuming that operator A in H is self-adjoint and independent of t . Multiplying (2.15) scalarly by du=dt , we get 1d 2 dt

 2     du    C kuk2 D f , du . A  dt  dt

For the right-hand side, we use the estimate  2   1  du  du  C 1 kf k2 .   f, dt 2  dt  2 We go to the inequality d kuk2  kuk2 C kf k2 , dt where kuk2

 2  du  2  D  dt  C kukA .

The desired a priori estimate   Z t 2 0 2 0 2 2 ku.t /k  exp.t / ku kA C kv k C exp. /kf . /k d 0

expresses stability with respect to the initial data and the right-hand side for the operator-differential equation (2.15).

24

Chapter 2 Stability of operator-difference schemes

2.2 Two-level schemes Two-level schemes are here used to solve numerically the Cauchy problem for an operator-differential equation of first order. The present study is based on the general theory of stability for operator-difference schemes. For operator-difference schemes, which are considered in finite-dimensional Hilbert spaces, we introduce key concepts of the stability theory. We formulate criteria for stability of two-level difference schemes with respect to the initial data along with typical estimates for stability with respect to the initial data and the right-hand side.

2.2.1 Key concepts For simplicity, we define a uniform grid in time as !N  D ! [ ¹T º D ¹t n D n ,

n D 0, 1, : : : , N0 ,

N0 D T º.

Denote by A, B : H ! H linear operators in H depending, in general, on  , t n . Consider the Cauchy problem for an operator-difference equation B.t n /

y nC1  y n C A.t n /y n D ' n , 

t n 2 ! ,

y 0 D u0 ,

(2.17) (2.18)

where y n D y.t n / 2 H is a desired function and ' n , u0 2 H are given. We use the index-free notation of the theory of difference schemes: y D y n , yO D y nC1 , yL D y n1 , y  yL yO  y ytN D , yt D .   Then equation (2.17) may be written as By t C Ay D ',

t 2 ! .

(2.19)

We define a two-level difference scheme as a set of the Cauchy problems (2.17), (2.18) that depend on the parameter  ; the notation (2.17) (2.18) (as well as (2.18), (2.19)) is called the canonical form of two-level schemes. For solvability of the Cauchy problem at a new time level, it is assumed that B 1 exists. Then equation (2.19) may be written as yO D Sy C  ', Q

S D E  B 1 A,

'Q D B 1 ',

(2.20)

where, as usual, E is the identity operator. The operator S is called the transition operator of the two-level scheme (the transition from a current time level to the next one).

25

Section 2.2 Two-level schemes

A two-level scheme is called stable if there exist positive constants m1 and m2 , independent of  , u0 , and ', such that for any u0 2 H , ' 2 H , t 2 !N  for the solution of (2.17), (2.18) the following estimate is valid: ky nC1 k  m1 ku0 k C m2 max k'. /k , 0t n

t n 2 ! ,

(2.21)

where k  k k  k are some norms. The inequality (2.21) reflects the continuous dependence of the solution of (2.17), (2.18) on the input data. The difference scheme y nC1  y n C A.t n /y n D 0,  y 0 D u0 B.t n /

t n 2 ! ,

(2.22) (2.23)

is called stable with respect to the initial data if for the solution of (2.22), (2.23), the following estimate holds: ky nC1 k  m1 ku0 k,

t n 2 ! .

(2.24)

The two-level difference scheme B.t n /

y nC1  y n C A.t n /y n D ' n , 

t n 2 ! ,

y0 D 0

(2.25) (2.26)

is called stable with respect to the right-hand side if the solution satisfies the inequality ky nC1 k  m2 max k'. /k , 0t n

t n 2 ! .

(2.27)

The difference scheme (2.22), (2.23) is said to be -stable (uniformly stable) with respect to the initial data in HD if there exist constants  > 0 and m1 , independent of  , n, such that for any n and all y n 2 H , the solution y nC1 of the difference equation (2.22) satisfies the estimate ky nC1 kD  ky n kD ,

t n 2 ! ,

(2.28)

and n  m1 . In the theory of difference schemes, one of the following quantities is selected as :  D 1,  D 1 C c ,  D exp .c /, where a constant c is independent of  , n.

c > 0,

26

Chapter 2 Stability of operator-difference schemes

In view of (2.20), rewrite equation (2.22) in the form y nC1 D Sy n .

(2.29)

The requirement of -stability is equivalent to fulfillment of the bilateral operator inequality D  DS  D (2.30) if DS is self-adjoint (DS D S  D). For an arbitrary operator of transition in (2.29), the condition of -stability is given by S  DS  2 D.

(2.31)

Let us formulate the discrete analog of Gronwall’s lemma. Lemma 2.2. From the estimate for the difference solution at the n C 1-st time level ky nC1 k  ky n k C  k' n k

(2.32)

it follows that the a priori estimate ky nC1 k  nC1 ky 0 k C

n X

nk k' k k

(2.33)

kD0

holds. Thus, from the levelwise estimate, we obtain an a priori estimate for the difference solution at any time level.

2.2.2 Stability with respect to the initial data Let us consider basic criteria for stability of two-level schemes with respect to the initial data [86]. Most important is the following theorem [125–127, 131] on the exact (coinciding necessary and sufficient) condition for stability in HA . Theorem 2.2. Assume that in equation .2.22/ A is a positive self-adjoint operator independent of n. The condition B

 A, t 2 ! 2

(2.34)

is necessary and sufficient for stability in HA , i.e., for the fulfillment of the estimate ky nC1 kA  ku0 kA , t 2 ! .

(2.35)

27

Section 2.2 Two-level schemes

Proof. Multiplying equation (2.22) scalarly by y t , we get .By t , y t / C .Ay, y t / D 0. Using the representation

(2.36)

1 1 O  y t , y D .y C y/ 2 2

rewrite (2.36) in the form   1   B  A y t , y t C .A.yO C y/, yO  y/ D 0. 2 2

(2.37)

For the self-adjoint operator A, we have .Ay, y/ O D .y, Ay/ O and .A.yO C y/, yO  y/ D .Ay, O y/ O  .Ay, y/. Substituting these relations into (2.37) and using the condition (2.34), we obtain the inequality (2.38) ky nC1 kA  ky n kA , which ensures the desired estimate (2.35). To prove the necessity of the inequality (2.35), assume that the scheme is stable in HA , i.e., the inequality (2.35) holds. We prove that this implies the operator inequality (2.34). Consider (2.37) at the initial time level n D 0:   y1  y0   . 2 B  A w, w C .Ay1 , y1 / D .Ay0 , y0 /, w D 2  By (2.35), this identity holds only if     B  A w, w  0. 2 Let y0 D u0 2 H be an arbitrary element, then the element w D B 1 Au0 2 H is arbitrary, too. Indeed, for any element w 2 H , we obtain u0 D A1 Bw 2 H since A1 exists. Thus, the inequality holds for all w 2 H , i.e., we have the operator inequality (2.34). The condition (2.34) is necessary and sufficient for stability not only in HA , but also in other norms. We now formulate (without proof) the stability result for HB (see [131, 134, 136] for more details). Theorem 2.3. Assume that in .2.22/, .2.23/ operators A and B are constant and B D B  > 0, A D A > 0.

(2.39)

Then the condition (2.34) is necessary and sufficient for stability of the scheme (2.22), (2.23) with respect to the initial data in HB with  D 1. Consideration of general time-dependent problems is based on using -stability.

28

Chapter 2 Stability of operator-difference schemes

Theorem 2.4. Let A and B be constant operators and A D A , B D B  > 0. Then the condition

1C 1 BA B (2.40)   is necessary and sufficient for -stability of the scheme .2.22/, .2.23/ in HB , i.e., for the fulfilment of ky nC1 kB  ky n kB . Proof. Writing (2.22) in the form of (2.29), we get from (2.30) the following condition for stability in HB : B  B  A  B. This bilateral operator inequality may be formulated in a more traditional representation using inequalities in the form of (2.40) for the scheme operators. We emphasize that in this theorem it is not assumed that the operator A is positive (or at least non-negative). Under the additional assumption on the positiveness of A, we get that the condition (2.40) is necessary and sufficient for the -stability of the scheme (2.22), (2.23) in HA . If   1, then stability, as in Theorem 2.2, is established for two-level difference schemes with the non-self-adjoint operator B. Theorem 2.5. Let A be a self-adjoint, positive, and constant operator. Then under the condition  A, (2.41) B 1C the scheme .2.22/, .2.23/ is -stable in HA . Proof. Adding and subtracting from the basic energy identity (see (2.37))     O y/ O  .Ay, y/ D 0 (2.42) 2 B  A y t , y t C .Ay, 2 the expression 1 .Ay t , y t /, 2 2 1C we get     1 2 2 B A y t , y t C .Ay,  .Ay t , y t / D 0. O y/ O  .Ay, y/  1C 1C In view of (2.41) and the self-adjointness of A, we obtain immediately .Ay, O y/ O  .Ay, y/ C .  1/.Ay, O y/  0.

29

Section 2.2 Two-level schemes

The inequality j.Ay, O y/j  kyk O A kykA with notation D

kyk O A kykA

yields the inequality 2  .  1/ C   0. It holds for all 1    , and so we go to the desired estimate kyk O A  kykA , which ensures stability in HA . Now we consider a priori estimates that express stability with respect to the righthand side. Such estimates are employed to study convergence of difference schemes for time-dependent problems.

2.2.3 Stability with respect to the right-hand side First, we show that stability with respect to the initial data in HR , R D R > 0 results in stability with respect to the right-hand side in the norm k'k D kB 1 'kR . Theorem 2.6. Assume that .2.17/, .2.18/ is -stable on HR with respect to the initial data, i.e., the estimate .2.28/ holds with ' n D 0. Then the scheme .2.17/, .2.18/ is stable with respect to the right-hand side and the following a priori estimate is true: ky nC1 kR  nC1 ku0 kR C

n X

nk kB 1 ' k kR .

(2.43)

kD0

Proof. As B 1 exists, we have that equation (2.17) may be written as y nC1 D Sy n C  'Q n ,

S D E  B 1 A,

'Q n D B 1 ' n .

(2.44)

From (2.44), we get ky nC1 kR  kSy n kR C  kB 1 ' n kR .

(2.45)

The requirement of -stability with respect to the initial data is equivalent to the boundedness of the norm of the transition operator S : kSy n kR  ky n kR ,

t 2 ! .

Because of this, from (2.45), we obtain ky nC1 kR  ky n kR C  kB 1 ' n kR .

30

Chapter 2 Stability of operator-difference schemes

Using the discrete analog of Gronwall’s lemma, we obtain the desired estimate (2.43), which expresses the stability of the scheme with respect to the initial data and the right-hand side. In particular, if R D A or R D B (under the condition A D A > 0 or B D > 0), then, from (2.43), we obtain elementary estimates for stability in the energy space HA or HB . Some new estimates for the two-level difference scheme (2.17), (2.18) can be obtained by coarsening the stability criterion (2.34). B

Theorem 2.7. Let A be a self-adjoint, positive, and constant operator and assume that B satisfies the condition 1C" A (2.46) B 2 with a constant " > 0 independent of  . Then the scheme (2.17), (2.18) satisfies the a priori estimate n 1C" X 2 2 2  ku0 kA C  k' k kB (2.47) ky nC1 kA 1 . 2" kD0

Proof. Multiplying equation (2.17) scalarly by 2y t , we obtain, similarly to (2.42), the energy identity     O y/ O D .Ay, y/ C 2 .', y t /. (2.48) 2 B  A y t , y t C .Ay, 2 The right-hand side of the above expression can be estimated as 2 .', y t /  2 k'kB 1 ky t kB  2 2  2 "1 ky t kB C k'kB 1 2"1 with a positive constant "1 . Substituting this estimate into (2.48), we get      2 2 .1  "1 /B  A y t , y t C .Ay, O y/ O  .Ay, y/ C k'kB 1 . 2 2"1 If the condition (2.46) holds, then it is possible to select "1 such that 1 D 1 C ", 1  "1 and so  1C" A/  0, .1  "1 /B  A D .1  "1 /.B  2 2 1C" 2  k'kB .Ay, O y/ O  .Ay, y/ C 1 . 2" The last inequality implies the estimate (2.47).

31

Section 2.2 Two-level schemes

Theorem 2.8. Let A be a self-adjoint, positive, and constant operator, and assume that B satisfies the condition  B  G C A, G D G  > 0. 2

(2.49)

Then .2.17/, .2.18/ satisfies the a priori estimate 2 ky nC1 kA



2 ku0 kA

n 1X C  k' k k2G 1 . 2

(2.50)

kD0

Proof. In the identity (2.48), we employ the estimate  2 .', y t /  2 .Gy t , y t / C .G 1 ', '/. 2 Substituting this estimate into (2.48) and taking into account (2.49), we get 1 .Ay, O y/ O  .Ay, y/ C  k'k2G 1 2 that, by a discrete analog of Gronwall’s lemma, gives (2.50). The convergence study of difference schemes is conducted in various classes of smoothness of the solution of the original differential problem, and therefore we must have a wide range of estimates. In particular, the right-hand side should be estimated in different and simply calculated norms. Only typical a priori estimates for solutions of operator-difference schemes were considered here.

2.2.4 Schemes with weights We now apply the above results to elementary schemes with weights for an operatordifferential equation of first order. The Cauchy problem du C Au D f .t /, dt

t > 0,

(2.51)

u.0/ D u0 ,

(2.52)

where A > 0 is associated with the two-level scheme with weights y nC1  y n C A.y nC1 C .1   /y n / D ' n ,  y 0 D u0 .

t n 2 ! ,

(2.53) (2.54)

The scheme (2.53), (2.54) may be written in the canonical form (2.19) with the operators B D E C  A, A > 0. (2.55)

32

Chapter 2 Stability of operator-difference schemes

Theorem 2.9. The scheme with weights .2.53/, .2.54/ is stable with respect to the initial data if and only if the following operator inequality holds:  1  (2.56) A A  0. A C   2 Proof. By A > 0, there exists A1 . Multiplying (2.53) by A1 , we go from (2.19), (2.55) to the scheme y BQ where

 yn Q n D 'Q n , C Ay 

nC1

BQ D A1 C  E,

t n 2 ! ,

AQ D E.

The necessary and sufficient condition for stability of this scheme with respect to the initial data in H D HAQ (Theorem 2.2) is formulated as the inequality   1 1 E  0. A C  2 Multiplying it from the left by A and from the right by A, we obtain (2.56). If   0.5, then the operator-difference scheme (2.53), (2.54) is unconditionally stable (stable for any  > 0).

2.3 Three-level schemes Three-level schemes are considered below using the reduction to equivalent two-level schemes. Estimates for stability with respect to the initial data and the right-hand side are obtained in various norms. Three-level schemes with weights are studied for an operator-differential equation of first order as well as for an elementary second-order equation.

2.3.1 Stability with respect to the initial data Considering stability of three-level difference schemes, we employ [129, 131, 134] the canonical form of three-level difference schemes: y nC1  y n1 C R.t n /.y nC1  2y n C y n1 / C A.t n /y n D ' n , 2 n D 1, 2, : : :

B.t n /

(2.57)

with a given y 0 D u0 ,

y 1 D v0.

(2.58)

Section 2.3 Three-level schemes

33

Let us obtain a condition for stability with respect to the initial data in the case of the constant (independent of n) self-adjoint operators A, B, and R, i.e., instead of the general scheme (2.57), we consider the scheme y nC1  y n1 (2.59) C R.y nC1  2y n C y n1 / C Ay n D 0. 2 Let us derive an elementary a priori estimate for the scheme (2.58), (2.59), which expresses stability with respect to the initial data. Suppose 1 un D .y n C y n1 /, w n D y n  y n1 (2.60) 2 and rewrite, using the identity 1 1 y n D .y nC1 C 2y n C y n1 /  .y nC1  2y n C y n1 / 4 4 the scheme (2.59) as B

w nC1 C w n 1 unC1 C un C R.w nC1  w n /  A.w nC1  w n / C A D 0. (2.61) 2 4 2 Multiply equation (2.61) scalarly by B

2.unC1  un / D w nC1 C w n , this yields the equality 1 .B.w nC1 C w n /, w nC1 C w n / C .R.w nC1  w n /, w nC1 C w n / 2 1  .A.w nC1  w n /, w nC1 C w n / 4 C .A.unC1 C un /, unC1  un / D 0.

(2.62)

For self-adjoint operators R and A and for a non-negative operator B .B  0/, it follows from (2.62) that (2.63) E nC1  E n , where, in view of the notation (2.60), we have 1 E nC1 D .A.y nC1 C y n /, y nC1 C y n / 4 C .R.y nC1  y n /, y nC1  y n / 1  .A.y nC1  y n /, y nC1  y n /. 4

(2.64)

Under certain constraints, the quantity E n , defined by (2.64), specifies a norm, and therefore the inequality (2.63) ensures stability of the operator-difference scheme with respect to the initial data. More accurately, the following statement is valid.

34

Chapter 2 Stability of operator-difference schemes

Theorem 2.10. Let the operators R and A in the operator-difference scheme .2.59/ be self-adjoint operators. Then, if the condition 1 B  0, A > 0, R > A 4

(2.65)

is fulfilled, the following a priori estimate holds: 1 1 nC1 2 2 C y n kA C ky nC1  y n k2R  ky nC1  y n kA ky 4 4 1 1 2 2  ky n C y n1 kA C ky n  y n1 k2R  ky n  y n1 kA , 4 4

(2.66)

i.e., the operator-difference scheme .2.59/ is stable with respect to the initial data. Considered three-level schemes demonstrate a complex structure of the norm (see (2.64)). In some important cases, on narrowing the class of difference schemes or on making stability conditions coarser, it is possible to use simpler norms [131,134,136].

2.3.2 Reduction to a two-level scheme To study multilevel difference schemes, it is convenient to reduce them to equivalent two-level schemes. In doing so, we obtain some fundamental results, in particular, a coinciding necessary and sufficient condition for stability. Denote [59] by H 2 the direct sum of spaces H : H 2 D H ˚ H . For vectors u D ¹u1 , u2 º, the operations of addition and multiplication in H 2 are defined in the coordinatewise manner, and the scalar product is .u, v/ D .u1 , v1 / C .u2 , v2 /. In H 2 , we define the operators (operator matrices)   G11 G12 GD G21 G22 such that the elements G˛ˇ are operators in H . A self-adjoint positive definite operator 2 , where the scalar product and the norm are G is associated with a Hilbert space HG given by .u, v/G D .Gu, v/, kukG D .Gu, u/1=2 . The three-level scheme (2.59) may be written as the two-level scheme B

y nC1  y n C Ay n D 0, 

n D 1, 2, : : :

with appropriately defined vectors y n , n D 1, 2, : : : .

(2.67)

35

Section 2.3 Three-level schemes

Therefore, for every n D 1, 2, : : : , we define the vector ² ³ 1 n n n1 n n1 /, y  y y D .y C y . 2

(2.68)

Under the conditions of Theorem 2.10, the above estimate (2.66) for stability with respect to the initial data may be expressed in this notation as ky nC1 kG  ky n kG ,

(2.69)

where

1 G22 D R  A. (2.70) 4 In view of (2.60), we can rewrite the two-level vector scheme (2.67), (2.68) as follows: unC1  un w nC1  w n (2.71) C B12 C A11 un C A12 w n D 0, B11   unC1  un w nC1  w n (2.72) C B22 C A21 un C A22 w n D 0. B21   The equality (2.71) is associated with a three-level scheme written in the form of (2.59). Taking into account the identities G11 D A,

G12 D G21 D 0,

unC1  un unC1 C un D un C , 2 2 2.unC1  un / D w nC1 C w n , we rewrite (2.59) in a more convenient form:  unC1  un w nC1  w n 1 unC1  un CR  A.w nC1  w n / C A C Aun D 0.   4 2  (2.73) To go from (2.71) to (2.73), we suppose that   B11 D B C A, B12 D R  A, A11 D A, A12 D 0. (2.74) 2 4 The equation (2.72) does not affect the three-level scheme (2.59). Considering the two-level schemes (2.67) with a self-adjoint operator A, we define  (2.75) B21 D Q, B22 D Q, A21 D 0, A22 D Q, 2 B

where Q is some self-adjoint positive operator. In the case of (2.74), (2.75), for the operators in (2.67), we have the representation  (2.76) B D A C Q, 2 where  (2.77) Q11 D B, Q12 D R  A, Q21 D Q, Q22 D 0. 4

36

Chapter 2 Stability of operator-difference schemes

Using this notation, under the conditions of Theorem 2.10, we can establish stability of the operator-difference scheme (2.59), i.e., the estimate (2.69), (2.70). The two-level vector scheme (2.67) with a self-adjoint positive operator A is stable with respect to 2 if and only if the initial data in HA  A. 2 In view of (2.76), the condition (2.78) is valid if B

(2.78)

Q  0. The latter condition is always fulfilled for an operator Q defined by (2.77) in the case of B  0 and 1 Q D R  A. (2.79) 4 2 In the case of (2.79), stability in HA refers to the case where the inequalities (2.69), (2.70) hold.

2.3.3 -stability of three-level schemes Assuming that a norm of the difference solution of a problem can both decrease and increase, we here consider -stable schemes [54], for which the condition for stability with respect to the initial data has the form ky nC1 kG  ky n kG ,

(2.80)

with  > 0. Theorem 2.11. Let the operators R and A in the difference scheme .2.59/ be selfadjoint operators. Then, under the condition  1 1 BC A  0, A > 0, R  A > 0 (2.81) 2C1 4 fulfilled with  > 1, the a priori estimate .2.80/, .2.70/ holds, i.e., the operatordifference scheme .2.59/ is -stable with respect to the initial data. Proof. The two-level vector difference scheme (2.67) is -stable with  > 1 in the case (see Theorem 4.5), where  A. (2.82) B C1 In view of (2.76), the inequality (2.82) may be rewritten as  1 QC A  0. (2.83) 2C1 Under the conditions of the theorem, the validity of the inequality (2.83) is checked immediately.

37

Section 2.3 Three-level schemes

Under more general conditions, -stability estimates of type (2.80) with arbitrary  > 0 can be obtained in the case of -dependent norms. In the operator-difference scheme (2.59), we introduce new unknowns y n D n z n , which yields B

z nC1  1 z n1 C R.z nC1  2z n C 1 z n1 / C Az n D 0. 2

(2.84)

We write the scheme (2.84) in the canonical form z BQ

nC1

 z n1 Q n D 0. Q nC1  2z n C z n1 / C Az C R.z 2

(2.85)

Direct calculations yield 2 C 1 B C  .2  1/R, BQ D 2 2 C 1 2  1 BC R, RQ D 4 2 2  1 B C .  1/2 R C A. AQ D 2

(2.86)

By Theorem 2.10, under the constraint BQ  0,

AQ > 0,

1 RQ  AQ > 0, 4

(2.87)

the scheme (2.85) is stable with respect to the initial data and the following estimate is valid: kznC1 kGQ  kzn kGQ , (2.88) where (see (2.68))

² zn D

³ 1 n .z C z n1 /, z n  z n1 . 2

Now we define the vector

²   ³ 1 1 n 1 n n1 n1 y Cz , z z . y D 2   n

(2.89)

Then the estimate (2.88) assumes the form ky nC1 kGQ  ky n kGQ ,

(2.90)

i.e., the original difference scheme (2.59) is -stable with respect to the initial data. The norm in (2.90) is defined by the operator GQ , for which Q GQ 11 D A,

GQ 12 D GQ 21 D 0,

1 Q GQ 22 D RQ  A. 4

Stability conditions may be formulated based on (2.86),(2.87).

(2.91)

38

Chapter 2 Stability of operator-difference schemes

Theorem 2.12. Let the operators B, R and A in the difference scheme .2.59/ be self-adjoint operators. Then, under the condition 2 C 1 B C  .2  1/R  0, 2 2  1 (2.92) B C .  1/2 R C A > 0, 2 2  1 B C . C 1/2 R  A > 0 2 fulfilled with  > 0, the a priori estimate (2.89)–(2.91) holds, i.e., the difference scheme (2.59) is -stable with respect to the initial data in H 2Q . G

2.3.4 Estimates in simpler norms The stability of the operator-difference schemes discussed above was established in Hilbert spaces with a complex composite norm (see (2.63), (2.64)). In considering of stability of three-level difference schemes, estimates are obtained [131, 134] for stability in simpler (compared to (2.64)) norms. The latter was achieved at the expense of stronger stability conditions. Let us formulate the result. Theorem 2.13. Let the operators R and A in the operator-difference scheme .2.59/ be self-adjoint operators. Then, under the constraint 1C" A 4 fulfilled with " > 0, the following a priori estimates are valid: B  0, A > 0, R >

1C" 2 C ky 1  y 0 k2R /, .ky 0 kA " 4 C 3" 2 2 C ky n  y n1 k2R  C ky 1  y 0 k2R /. .ky 0 kA ky nC1 kA " Proof. In the index-free notation 2 2 ky nC1 kA

y n D y,

y nC1 D y, O

(2.93)

(2.94) (2.95)

yO  y D y t ,

and, for E nC1 defined by (2.64), we have 1 2 E nC1 D .A.yO C y/, yO C y/ C  2 .Ry t , y t /  .Ay t , y t / 4 4 D .Ay, O y/ C  2 .Ry t , y t /. Substitution of yO D y C y t into (2.96) yields E nC1 D .Ay, y/ C  .Ay, y t / C  2 .Ry t , y t / 2  kykA C  kykA ky t kA C  2 ky t k2R .

(2.96)

39

Section 2.3 Three-level schemes

Taking into account the third inequality in (2.93), we get 2 kykA ky t kR C  2 ky t k2R .1 C "/1=2 2  2.kykA C  2 ky t k2R /.

2 E nC1  kykA C

Thus, we have established a lower estimate for the composite norm 2 E nC1  2.ky n kA C ky nC1  y n k2R /.

(2.97)

An upper estimate can be established in a similar manner. In (2.96), we suppose that y D D yO  y t ; then, in view of (2.93), we obtain: E nC1 D .Ay, O y/ O   .Ay, O y t / C  2 .Ry t , y t / 2  kyk O A   kyk O A ky t kA C  2 ky t k2R 2 2  kyk O A  kyk O A ky t kR C  2 ky t k2R . .1 C "/1=2

For an arbitrary ˇ > 0, we have  2 O A C 1 E nC1  .1  ˇ/kyk

 1  2 ky t k2R . ˇ.1 C "/

(2.98)

Assume that ˇ D 1=.1 C "/, then from (2.98) we get E nC1 

" 2 . ky nC1 kA 1C"

(2.99)

Taking into account (2.97) and (2.99), the stability estimate (2.63) yields the desired stability estimate (see (2.94)) for the three-level scheme (2.59) in HA . To prove the estimate (2.95), we put ˇ D .1 C "/1=2 , so that 1ˇ D

" .1 C "/1=2  1 D . 1=2 .1 C "/ 1 C " C .1 C "/1=2

In view of .1 C "/1=2 < 1 C 0.5", we go from (2.98) to the second lower estimate in the composite norm: 2" 2 C ky nC1  y n k2R /. (2.100) .ky nC1 kA E nC1 > 4 C 3" The inequality (2.63) and the estimates (2.97),(2.100) yield the estimate (2.95). Estimates of type (2.94) are natural for three-level schemes arising from evolutionary equations of the first order in time (a parabolic equation of the second order in space); estimates of type (2.95) are typical for equations of the second order in time (a hyperbolic equation of the second order in space).

40

Chapter 2 Stability of operator-difference schemes

2.3.5 Stability with respect to the right-hand side Let us provide some elementary estimates for stability with respect to the initial data and the right-hand side for three-level schemes. Instead of (2.59), now we consider the scheme B

y nC1  y n1 C R.y nC1  2y n C y n1 / C Ay n D ' n . 2

(2.101)

Theorem 2.14. Let the operators R and A in .2.101/ be self-adjoint operators. Then, under the condition 1 (2.102) B  "E, A > 0, R > A 4 fulfilled with a constant " > 0, the difference solution satisfies the a priori estimate E nC1  E 1 C

n 1 X  k' k k2 , 2"

(2.103)

kD1

E nC1

n 1X 2  E1 C  k' k kB 1 . 2

(2.104)

kD1

Proof. Similarly to the proof of Theorem 2.10 (see (2.62)), we obtain the equality 1 .B.w nC1 C w n /, w nC1 C w n / C E nC1 D .' n , w nC1 C w n / C E n . 2 To derive the estimate (2.103) with " > 0, under the condition (2.102), we employ the inequality 1  "kw nC1 C w n k2 C k' n k2 . .' n , w nC1 C w n /  2 2" The inequality 1  2 2 .' n , w nC1 C w n /  C k' n kB kw nC1 C w n kB 1 . 2 2 is used to check that the estimate (2.104) holds. Another stability estimate for three-level schemes (2.101) with respect to the righthand side can be obtained (see [131, 134]) on the basis of the estimates (2.94), (2.95) with somewhat stronger constraints imposed on R.

2.3.6 Schemes with weights for equations of first order For numerically solving the problem du C Au D f .t /, dt u.0/ D u0

t > 0,

41

Section 2.3 Three-level schemes

along with two-level schemes employing a single weight parameter, the three-level scheme with weights is often used: y nC1  y n1 C A.1 y nC1 C .1  1  2 /y n C 2 y n1 / D ' n , 2 n D 1, 2, : : : ,

(2.105)

y 0 D u0 ,

(2.106)

y 1 D u1 .

To specify the second initial condition (u1 in (2.106)) with the second-order approximation, we employ in the simplest case the two-level scheme y1 C y0 y1  y0 CA D ' 0.  2 We write the scheme (2.105), (2.106) in the canonical form (2.57) with 1 C 2 B D E C  .1  2 /A, R D A. 2

(2.107)

Consider the case with a variable positive operator A. Theorem 2.15. Assume that A > 0 and the condition 1 1  2 , 1 C 2 > , 2

(2.108)

holds, then the scheme .2.105/, .2.106/ is stable with respect to the initial data and the difference solution (for ' D 0) satisfies the estimate ky nC1 k  ky 1 k , where

(2.109)

  1 1 1 ky nC1 k2 D ky nC1 C y n k2 C 1 C 2  ky nC1  y n k2 . 4 2 2

Proof. Acting on (2.57), (2.107) with the operator A1 , we obtain: y BQ

nC1

 y n1 Q n D 'Q n , Q nC1  2y n C y n1 / C CAy C R.y 2

n D 1, 2, : : : , (2.110)

where 1 C 2 RQ D E, AQ D E, 'Q D A1 '. (2.111) 2 Let us apply Theorem 2.10 to the scheme (2.110), (2.111). Under the assumptions (2.108), the following inequalities are valid:   1 C 2 1 2 1 Q Q   E > 0, R AD 4 2 4 BQ D A1 C  .1  2 /E,

BQ D A1 C .1  2 /E  0,

42

Chapter 2 Stability of operator-difference schemes

which ensure stability with respect to the initial data, and, for the difference solution of the problem with the homogeneous right-hand side (with ' D 0), the estimate (2.109) holds. A wide range of a priori estimates for stability with respect to the right-hand side for the three-level scheme with weights (2.57), (2.58) can be found in [131, 134, 136].

2.3.7 Schemes with weights for equations of second order Let us consider conditions for stability of schemes with weights for the Cauchy problem d 2u C Au D f .t /, t > 0, dt 2 du .0/ D v 0 u.0/ D u0 , dt

(2.112) (2.113)

with a constant, self-adjoint, and positive operator A. To solve numerically the problem (2.112), (2.113), it is natural to use the scheme with weights y nC1  2y n C y n1 C A.1 y nC1 C .1  1  2 /y n C 2 y n1 / D ' n , 2 (2.114) n D 1, 2, : : : , y 0 D u0 ,

y 1 D u1 .

(2.115)

The scheme (2.114), (2.115) may be written in the canonical form (2.57), (2.58) with the operators B D  .1  2 /A,

RD

1 1 C 2 EC A. 2  2

(2.116)

Theorem 2.16. Assume that the operator A is constant and A D A > 0, and the condition 1 (2.117) 1  2 , 1 C 2  2 is fulfilled, then the scheme .2.114/, .2.115/ is stable with respect to the initial data and the difference solution (for ' D 0) satisfies the estimate ky nC1 k  ky1 k , where now

1 2 C ky n  y n1 k2R 1 A . ky nC1 k2 D ky nC1 C y n kA 4 4

Proof. By Theorem 2.10, the proof is trivial.

43

Section 2.4 Stability in finite-dimensional Banach spaces

Using additional information about the norm of A, the stability condition (2.117) can be reduced to weaker formulations. Taking into account that A  kAkE, the stability condition (2.65) for the scheme (2.57), (2.116) holds with 1 2 . 1  2 , 1 C 2   2 2  kAk Thus, in particular, we obtain the stability conditions for the explicit scheme (1 D 2 D 0 in (2.114)).

2.4 Stability in finite-dimensional Banach spaces A study of methods for solving time-dependent problems is often performed in Banach spaces. Particular attention is paid to a formulation of sufficient conditions for stability in the spaces L1 .!/ and L1 .!/. In the theory of difference schemes, such a study is based on applying the maximum principle for grid equations. In our investigations, we use the concept of the logarithmic norm for the corresponding operators in finitedimensional Banach spaces. As an example, two-level schemes with weights will be analyzed for the numerical solving of a boundary value problem for a one-dimensional parabolic equation.

2.4.1 The Cauchy problem for a system of ordinary differential equations Consider a system of linear ODEs of first order: X dui aij .t /uj D fi .t /, C dt m

t > 0,

i D 1, 2, : : : , m.

(2.118)

j D1

Assume that u D u.t / D ¹u1 , u2 , : : : , um º, A D ¹aij º, then we can write (2.118) in matrix (operator) form as du C A.t /u D f .t /. (2.119) dt The equation (2.119) is supplemented with the initial condition u.0/ D u0 ,

u0 D ¹u01 , u02 , : : : , u0m º.

(2.120)

In computational practice, the stability of the difference solution of the problem (2.119), (2.120) in L1 (in C ) and in L1 is of great interest. We recall some basic concepts of linear algebra. For a norm of a vector and a norm of a matrix, consistent with it in L1 , we have kwk1 D max jwi j, 1im

kAk1 D max

1im

m X j D1

jaij j.

(2.121)

44

Chapter 2 Stability of operator-difference schemes

Similarly, in L1 , we obtain kwk1 D

m X

jwi j,

kAk1 D max

1j m

iD1

m X

jaij j.

(2.122)

iD1

The problem (2.119), (2.120) will be considered under the following constraints. Assume that the diagonal elements of the matrix A are non-negative, and there is rowwise or column-wise diagonal dominance, i.e., we have ai i 

m X

jaij j,

i D 1, 2, : : : , m

(2.123)

j D 1, 2, : : : , m

(2.124)

i6Dj D1

(weak diagonal dominance by rows) or ajj 

m X

jaij j,

j 6DiD1

(weak diagonal dominance by columns). The logarithmic norm of the matrix A is defined [25, 57] by the number kE C ıAk  1 . ı ı!0C

ŒA D lim

For the logarithmic norm of a matrix in L1 (consistent with (2.121)) and in L1 (consistent with (2.122)), we have the expressions   m X 1 ŒA D max ai i C jaij j , 1im

i6Dj D1

  m X jaij j . 1 ŒA D max ajj C 1j m

j 6DiD1

In view of the restrictions (2.123), (2.124), we have that the logarithmic norm of the matrix A in the Cauchy problem (2.119), (2.120) satisfies ŒA  0

(2.125)

in the corresponding space (in L1 with (2.123) and in L1 with (2.124)). Among the properties of the logarithmic norm (see [25, 26]), we highlight the following: (1) ŒcA D cŒA,

c D const  0;

(2) ŒcE C A D c C ŒA,

c D const ;

(3) kAwk  max¹ŒA, ŒAº kwk.

45

Section 2.4 Stability in finite-dimensional Banach spaces

The emphasis is placed on property (3), which allows to easily get a lower bound of the norm Aw. This bound can be combined with the standard upper bound of Aw: kAwk  kAk kwk. Using the logarithmic norm, it is easy to obtain a priori estimates for systems of linear and nonlinear ODEs. For example, for the problem (2.119), (2.120) with the constant operator A, we have the estimate Z t exp.ŒA /kf . /kd /, ku.t /k  exp.ŒAt /.ku0 k C 0

which expresses stability with respect to the initial data and the right-hand side.

2.4.2 Scheme with weights Let us study the stability of difference schemes for the problem (2.119), (2.120). We denote the approximate solution at time level t n by y n , and write a two-level difference scheme with weights y nC1  y n C A.y nC1 C .1   /y n / D ' n , 

(2.126)

where, e.g., A D A. t nC1 C .1   /t n / with the initial data y 0 D u0 .

(2.127)

A sufficient condition for stability of the scheme (2.126), (2.127) is formulated as the following statement. Theorem 2.17. Assume that in the Cauchy problem .2.119/, .2.120/ the matrix A satisfies the restriction .2.123/ (or .2.124/). Then the difference scheme with weights .2.126/, .2.127/ is unconditionally stable with  D 1 and is conditionally stable with 0   < 1 in L1 (in L1 ), if and only if  1 1 . (2.128) max ai i  1   1im In this case, the difference solution satisfies the a priori estimate ky

nC1

k  ku k C 0

n X

 k' k k.

(2.129)

kD0

Proof. From (2.126), it follows that .E C  A/y nC1 D .E  .1   /A/y n C  ' n , and therefore k.E C  A/y nC1 k  k.E  .1   /A/y n k C  k' n k.

(2.130)

46

Chapter 2 Stability of operator-difference schemes

For the left-hand side of (2.130), by the above-mentioned properties of the logarithmic norm and in view of (2.125), we have k.E C  A/y nC1 k  ŒE   A ky nC1 k D .1    ŒA/ky nC1 k  ky nC1 k. For the first term in the right-hand side of (2.130), we obtain k.E  .1   /A/y n k  kE  .1   /Ak ky n k. We investigate this estimate in more detail for L1 . The case L1 is studied in a similar manner. Considering (2.121) and taking into account the condition of diagonal dominance (2.123), we have m ˇ ˇ X ˇ ˇ aij /ˇ kE  .1   /Ak D max ˇ1  .1   / .ai i C 1im

i6Dj D1

m X

 max .j1  .1   / ai i j C .1   / 1im

jaij j/

i6Dj D1

 max .j1  .1   / ai i j C .1   / ai i /  1 1im

with 0    1 and under the restriction (2.128) for the time step. The substitution into (2.130) yields the inequality ky nC1 k  ky n k C  k' n k,

(2.131)

which immediately implies the desired estimate (2.129) for stability with respect to the right-hand side and the initial data. It is often suitable to obtain the condition for stability in L1 using the corresponding results for stability in the dual space L1 (see [47]). The fully implicit scheme (with  D 1) is an unconditionally stable scheme. Any scheme with   1 indicates the same property. The scheme with weights (2.126), (2.127) may be treated (P. Matus called our attention to this fact) as the fully implicit scheme yQ nC1  y n C AyQ nC1 D ' n  for the function yQ nC1 D y nC1 C .1   /y n . Similarly to the proof of Theorem 2.17, we have .1 C  ŒA/kyQ nC1 k  ky n k C   k' n k. Under the restriction   1, it follows that kyQ nC1 k   ky nC1 k  .  1/ky n k and the inequality (2.131) holds.

Section 2.5 Stability of projection-difference schemes

47

2.4.3 Difference schemes for a one-dimensional parabolic equation To illustrate conditions for stability in Banach spaces, consider a model problem   @ @u @u  k.x/ D f .x, t /, 0 < x < l, t > 0, @t @x @x u.0, t / D 0,

u.l, t / D 0,

t > 0,

u.x, 0/ D u0 .x/, 0 < x < l under the natural assumption that k.x/ > 0, 0 < x < l. On a uniform grid !N with spacing h, we consider the scheme with weights (2.126), (2.127), which, e.g., at the interior points (x 2 !) satisfies Ay D .ayxN /x 1 y.x C h/  y.x/ 1 y.x/  y.x  h/ D  k.x C 0.5h/ C k.x  0.5h/ . h h h h It is easy to see that, for A, we have ŒA  0 in L1 , and, for the scheme (2.126), (2.127), the stability condition (2.128) seems like this: 1 h2   . max k.x ˙ 0.5h/ 1   x2! The difference solution satisfies the following estimate for stability in the uniform norm: n X  k' k k1 . ky nC1 k1  ku0 k1 C kD0

In a similar way, we can investigate more general boundary value problems for parabolic equations.

2.5 Stability of projection-difference schemes General conditions for stability of projection-difference schemes (finite element procedures) are formulated here for numerically solving linear time-dependent problems in the form of inequalities for the corresponding bilinear forms. A -stability condition is considered for such schemes with an arbitrary . Estimates are presented for stability with respect to the initial data and the right-hand side in various norms. The stability of three-level schemes with weights is investigated for an evolutionary equation of first order. Some other results of the general theory of stability of operator-difference schemes can be adapted to finite element procedures, too.

48

Chapter 2 Stability of operator-difference schemes

2.5.1 Preliminary observations The use of finite element procedures for solving time-dependent problems of mathematical physics is reflected in basic manuals on numerical analysis [70, 88, 107]. To justify finite element techniques, the emphasis is on obtaining a priori estimates for stability with respect to the initial data and the right-hand side for the approximate solution. The fundamental issue is to study the solution dependence on the parameters of discretization in space and in time. Such stability estimates provide the basis for investigating the convergence of the approximate solution to the exact one. Nowadays, stability estimates for finite element procedures are obtained separately and independently for each particular technique. There was a similar situation in the theory of difference schemes until the appearance of the general theory of difference schemes developed by A. Samarskii. This theory formulates the general necessary and sufficient condition for stability of two- and three-level difference schemes, which are considered in Hilbert spaces of grid functions. In this theory, a difference scheme is written in a single canonical form, and a necessary and sufficient condition is formulated in terms of operator inequalities. Using this theoretical framework, it is possible to analyze all basic classes of difference schemes for typical problems of mathematical physics. To study a particular scheme, we only need, in this approach, to write it in the canonical form and to check the general condition for stability. It seems natural to construct a similar stability theory for finite element procedures. As expected, this theory should be very close to the stability theory for finite difference schemes. It should be noted that we have two possibilities for these investigations. The first possibility is associated with constructing the corresponding difference schemes for coefficients of the expansion of the discrete solution over finite element basis functions. Treating these schemes as operator equations in a Hilbert space of vectors with the usual Euclidean norm, we can employ the results of stability of difference schemes. The second approach to construct a general theory of stability takes into account in a more precise way specific features of finite element techniques. It is based on the consideration of the original finite element procedures in a projection formulation. In doing so, the basic results can be obtained without the explicit consideration of matrix problems for coefficients of a finite element expansion, just involving properties of bilinear forms. Below, on the basis of the works [145, 209], we construct the stability theory for two-level finite element techniques. A sufficient condition for stability with respect to the initial data is derived for the techniques written in the canonical form. A priori estimates for stability with respect to the right-hand side are also presented. A coinciding necessary and sufficient condition is obtained for finite element procedures with symmetric bilinear forms. As an example, a two-level technique with weights is considered for a parabolic equation.

49

Section 2.5 Stability of projection-difference schemes

2.5.2 Stability of finite element techniques The stability theory for difference schemes is based on general methodological principles. The same framework is also used here for finite element techniques. The primary principles are formulated as follows: 

a finite element procedure (projection-difference scheme) is treated as a separate object of study, which is formally independent of the original differential problem;



finite element procedures are considered in subspaces of finite elements using a unified (canonical) form;



stability conditions are formulated as inequalities for the corresponding bilinear forms.

Using these principles, it is possible to obtain general and rather simple conditions which allow to investigate stability of finite element procedures in a single manner. In doing so, we can conduct both a classification of the well-known techniques and a stability analysis for new procedures. It is natural to consider finite element procedures in a generalized formulation that is based on using bilinear forms without any reduction to operator formulations. The distinctive feature of finite difference schemes is that they are just oriented to the employment of operator formulations. Let us solve numerically an initial-boundary value problem in a domain  with a boundary @. Let ., /, k  k be the scalar product and the norm in L2 ./, respectively, such that Z .u, v/ D

u.x/v.x/dx, 

kuk D .u, u/1=2 .

A symmetric positive definite bilinear form d.u, v/ such that d.u, v/ D d.v, u/,

d.u, u/  ıkuk2 ,

ı > 0,

is associated with the Hilbert space Hd equipped with the following scalar product and norm: .u, v/d D d.u, v/, kukd D .d.u, u//1=2 . Suppose t D t n D nt , n D 0, 1, : : :, where  > 0 is a constant time step. The upper index n is used to denote unknowns at the n-th time level t n . A finite-dimensional space of finite elements is denoted by V h , and y n .y n 2 V h / stands for the approximate solution at the time level t D t n . Finite element procedures for time-dependent problems are constructed using elementary discretizations in time. In a two-level technique, the solution at a new time level .n C 1/ is evaluated from the solution at the previous time level, i.e., calculations involve y n and y nC1 .

50

Chapter 2 Stability of operator-difference schemes

Similarly to finite difference schemes, we can write a finite element procedure as   nC1  yn n y , v C an .y n , v/ D .f n , v/, 8v 2 V h , n D 1, 2, : : : , (2.132) b  where b n ., /, an ., / stand for some real bilinear forms. For a given y 0 , we seek the approximate solution y n , n D 1, 2, : : : of equation (2.132). The initial data for (2.132) can be specified, e.g., like this: .y 0 , v/ D .u0 , v/,

8v 2 V h ,

where u0 .x/ 2 V h defines the initial condition for the exact solution. Any finite element procedure may be written in the form (2.132), which is called the canonical form of two-level projection-difference schemes. It is natural to analyze stability of finite element procedures with respect to the initial data separately from considering the right-hand side. As a rule, when speaking of stability, we mean stability with respect to the initial data. That is why we start with a homogeneous (f n D 0 in (2.132)) two-level scheme   nC1  yn n y (2.133) , v C an .y n , v/ D 0, 8v 2 V h , n D 1, 2, : : : b  with a given y0 2 V h . The projection-difference scheme (2.133) is said to be stable (more precisely, stable with respect to the initial data) in a space Hd if, for any y 0 2 V h , the approximate solution satisfies the estimate ky nC1 kd  ky n kd .

(2.134)

The stability condition (2.134) is not always reasonable. For example, a norm of the exact solution of the original differential problem may both decrease and increase with some rate. This situation is typical for ill-posed evolutionary problems, where a solution norm may increase and we need to introduce a regularization in order to control this growth. It is natural to try to obtain the approximate solution that is consistent with the exact solution in the sense of behavior in time. For this purpose, instead of (2.134), we introduce a more general concept of -stability. The projection-difference scheme (2.133) is called a -stable scheme if the approximate solution satisfies the estimate ky nC1 kd  ky n kd , with  > 0. If  > 1, then we usually use the following two representations:  D exp.c /,

 D 1 C c ,

(2.135)

51

Section 2.5 Stability of projection-difference schemes

where a positive constant c is independent of  . For the above , from (2.135), it follows that the following estimate for stability with respect to the initial data: ky nC1 kd  exp.ct nC1 /ky n kd

(2.136)

holds. If 0 <  < 1, then the estimate (2.136) holds for  D exp.c / with a negative constant c. In the stability theory of finite difference schemes, the exact (coinciding necessary and sufficient) conditions for their stability are established. They will serve us as a natural reference point for obtaining conditions for stability of projection-difference schemes. In fact, we will conduct a direct transfer of the results derived for finite difference schemes onto a class of finite element procedures.

2.5.3 Stability of projection-difference schemes It should be noted that an adequate mathematical tool of the stability theory for finite difference schemes is associated with operator formulations in the corresponding finite-dimensional Hilbert spaces. Namely, a finite difference scheme is written as an operator equation and stability conditions are formulated in the form of operator inequalities. For finite element procedures, it is natural to restrict ourselves to considering the corresponding bilinear forms. Any reduction to operator formulations seems to be redundant. This fact is reflected, in particular, in the choice of the canonical form of projection-difference schemes in the form (2.132). At the beginning assume that, in the scheme (2.133), the bilinear forms b., / and a., / are constant, i.e., they are independent of n:   nC1  yn y (2.137) , v C a.y n , v/ D 0, 8v 2 V h , n D 1, 2, : : : . b  Let us start with obtaining the energy identity for the scheme (2.137). We use the index-free notation, where y D y n,

yO D y nC1 ,

yt D

y nC1  y n . 

Lemma 2.3. Let in the projection-difference scheme .2.137/ the bilinear form a., / be symmetric. Then the following identity holds:    O y/ O  a.y, y/ D 0, n D 1, 2, : : : . (2.138) 2 b.y t , y t /  a.y t , y t / C a.y, 2 Proof. To prove (2.138), we use the above notation and the fact that 1 1 O  y t , y D .y C y/ 2 2

52

Chapter 2 Stability of operator-difference schemes

then we can rewrite (2.137) in the form  1 b.y t , v/  a.y t , v/ C a.yO C y, v/ D 0, 2 2 8v 2 V h , n D 1, 2, : : : .

(2.139)

Selecting v D 2y t D 2.yO  y/ and taking into account the symmetric property of the bilinear form a., /, from (2.139), we obtain immediately the identity (2.138). A similar identity holds in a more general case of variable (depending on n) bilinear forms. The main result on the stability of the projection-difference scheme (2.137) in Ha is formulated as follows. Theorem 2.18. Let in the scheme .2.137/ the bilinear form a., / be symmetric and positive definite. Then the condition  (2.140) b.v, v/  a.v, v/, 8v 2 V h 2 is necessary and sufficient for the a priori estimate a.y nC1 , y nC1 /  a.y n , y n /, n D 1, 2, : : : ,

(2.141)

i.e., the projection-difference scheme .2.137/ is stable in Ha . Proof. We use the identity (2.138). Under the condition (2.140), from the identity (2.138), we obtain (2.141), and therefore the condition (2.140) is sufficient. To prove the necessity of this condition, we consider the estimate (2.141) with n D 0. In view of the identity (2.138), we have  (2.142) b.y t , y t /  a.y t , y t /  0, n D 0. 2 Any function v from V h may be represented as .y 1  y 0 /= , choosing an appropriate initial condition y 0 2 V h . Therefore, it follows from (2.142) that (2.140) holds. Thus, the condition (2.140) is necessary for the stability of (2.137) in Ha . The resulting stability conditions provide stability of projection-difference schemes in various norms. As a typical example, we mention the stability of the scheme (2.137) in Hb under the additional assumption that the bilinear form b., / is symmetric. Namely, the following statement is true. Theorem 2.19. Let in the scheme .2.137/ the bilinear forms b., / and a., / be symmetric and positive definite. Then, if the inequality .2.140/ is valid, the estimate b.y nC1 , y nC1 /  b.y n , y n /, n D 1, 2, : : : , holds, i.e., the projection-difference scheme .2.137/ is stable in Hb .

(2.143)

53

Section 2.5 Stability of projection-difference schemes

Proof. It is sufficient to employ the second main energy identity for the scheme (2.137). Lemma 2.4. Let in the projection-difference scheme .2.137/ the bilinear forms b., / and a., / be symmetric. Then the following identity holds:     O y/ O  b.y, y/ C a.yO C y, yO C y/ D 0. (2.144)  2 b.y t , y t /  a.y t , y t / C b.y, 2 2 Proof. To check (2.144), assume that in (2.137) v D 2 yO D  2 y t C  .yO C y/. In view of the symmetry of the bilinear form b., /, we get O y/ O  b.y, y/. b.y t , v/ D  2 b.y t , y t / C b.y, Similarly, we have  3 a.yO C y, yO C y/  a.y t , y t /. 2 2 Substitution of these expressions into (2.137) leads to the required identity (2.144). a.y, v/ D

Under the conditions of the theorem, it follows immediately from (2.144) that the stability estimate (2.143) is valid for the scheme (2.137) in Hb .

2.5.4 Conditions for -stability of projection-difference schemes Special attention should be given to -stability conditions for finite element techniques, which can significantly extend the class of the schemes investigated here. We consider separately stability of projection-difference schemes with  > 1 (a norm of the approximate solution increases in time) and schemes with 0 <  < 1 (a norm decreases in time). As in the proof of Theorem 2.19, in the following we restrict ourselves to formulating a sufficient condition for stability only. For schemes with  > 1, our study is performed without the assumption that the bilinear form b., / is symmetric. Theorem 2.20. Let in .2.137/ the bilinear form a., / be symmetric and positive definite. Then, under the condition  (2.145) a.v, v/, 8v 2 V h , b.v, v/  1C fulfilled with  > 1, the a priori estimate a.y nC1 , y nC1 /  2 a.y n , y n /, n D 1, 2, : : : ,

(2.146)

holds, and therefore the projection-difference scheme .2.137/ is -stable in the space Ha .

54

Chapter 2 Stability of operator-difference schemes

Proof. We start from the basic energy identity (2.138). Adding and subtracting the expression 1 a.y t , y t /, 2 2 1C we rewrite (2.138) in the form    1 2 O y/ O  a.y, y/  a.y t , y t / C a.y,  a.y t , y t / D 0. 2 b.y t , y t /  1C 1C In view of (2.145), we obtain the inequality . C 1/a.y, O y/ O  . C 1/a.y, y/  .  1/a.yO  y, yO  y/  0. The bilinear form a., / is symmetric and therefore we have a.y, O y/ O  a.y, y/ C .  1/a.y, O y/  0. Taking into account ka.y, O y/k  .a.y, O y/a.y, O y//1=2 , we obtain the inequality a.y, O y/ O  a.y, y/  .  1/ .a.y, O y/a.y, O y//1=2  0. It holds with a.y, O y/ O  2 a.y, y/, i.e., under the condition that the a priori estimate for -stability (2.146) is fulfilled for the scheme (2.137) in Ha . It should be highlighted that -stability with  > 1 was proved without the assumption that the bilinear form b., / is symmetric. In a more general case, -stability of schemes with an arbitrary  > 0 is established using the assumption that the bilinear forms b., / and a., / are symmetric forms. Namely, the following fact is valid. Theorem 2.21. Let in the scheme .2.137/ the bilinear forms b., / and a., / be symmetric and positive definite. Assume that  > 0 and the bilateral inequality 1C 1 b.v, v/  a.v, v/  b.v, v/, 8v 2 V h ,  

(2.147)

holds. Then the projection-difference scheme .2.137/ is -stable in Hb , i.e., the approximate solution satisfies the estimate b.y nC1 , y nC1 /  2 b.y n , y n /, n D 1, 2, : : : .

(2.148)

Proof. Using the notation y n D n w n , the inequality (2.148) is equivalent to b.w nC1 , w nC1 /  b.w n , w n /,

(2.149)

55

Section 2.5 Stability of projection-difference schemes

i.e., it is sufficient to prove the standard ( D 1) stability for the approximate solution w n 2 V h , n D 0, 1, : : : . The projection-difference scheme (2.137) may be written in the form   nC1  wn w (2.150) , v C a.w Q n , v/ D 0, 8v 2 V h , n D 1, 2, : : : , b  where we use the symmetric bilinear form 1 1 a.w, Q v/ D b.w, v/ C a.w, v/,  

8w, v 2 V h .

The stability conditions for the scheme (2.150) were discussed above (see Theorem 2.19). The estimate (2.149) will take place if a.v, Q v/ > 0,

8v 2 V h ,

(2.151)

 (2.152) a.v, Q v/, 8v 2 V h . 2 The inequality (2.151) (see proof of Theorem 2.19) can be replaced by the weak inequality (2.153) a.v, Q v/  0, 8v 2 V h , b.v, v/ 

i.e., we only require non-negativity of the bilinear form a., Q /. The inequality (2.153) is just the left inequality of (2.147), whereas (2.152) is its right part. The validity of (2.149) implies that the desired estimate (2.148) for -stability holds for the scheme (2.137) in Hb .

2.5.5 Schemes with weights Let us now consider two-level schemes with weights as a typical example of projection-difference schemes. We formulate conditions for their stability in the form of the corresponding inequalities for bilinear forms. The focus is on schemes with nonsymmetric bilinear forms. At the beginning, we consider a scheme with a symmetric form a., /. We study the projection-difference scheme   nC1  yn y (2.154) , v C a.y nC1 C .1   /y n , v/ D 0, b  8v 2 V h ,

n D 1, 2, : : : ,

where  is a number (weight). If  D 0, then the scheme (2.154) is an explicit (forward-time) scheme; for  D 1, we obtain a fully implicit (backward-time) scheme; and  D 0.5 yields a symmetric (the so-called Crank–Nicolson) scheme. Assume that the bilinear form a., / is symmetric. Then, on the basis of Theorem 2.18, we obtain the following result.

56

Chapter 2 Stability of operator-difference schemes

Theorem 2.22. Let in the scheme with weights .2.154/ the bilinear form a., / be symmetric and positive definite. Then the condition  1 (2.155)  a.v, v/  0, 8v 2 V h b.v, v/ C   2 is necessary and sufficient for the stability of the scheme (2.154) in the space Ha . Proof. To establish this statement, we write the scheme with weights (2.154) in the canonical form   nC1  yn Qb y (2.156) , v C a.y Q n , v/ D 0,  8v 2 V h ,

n D 1, 2, : : :

with the bilinear forms Q b.w, v/ D b.w, v/ C   a.w, v/,

a.w, Q v/ D a.w, v/,

8w, v 2 V h .

The necessary and sufficient condition for the stability of the scheme (2.156) Q v/   a.v, b.v, Q v/, 2

8v 2 V h

leads to the condition (2.155). From (2.155), it follows that for b.v, v/  0 the condition   0.5 is sufficient for the stability of the scheme (2.154). If the inequality a.v, v/  b.v, v/,

8v 2 V h

holds, then stability takes place for  < 0.5 under the following restrictions on a time step: 1   0 D .

.0.5   / We have a typical situation with D .h/, i.e., in this case, we are talking about conditional stability of schemes with weights. In applied mathematical modeling, the scheme with weights (2.154) very often demonstrate the peculiarities such that the bilinear form a., / is nonsymmetric whereas the bilinear form b., / is symmetric. Therefore, there is no possibility to directly use the above results on stability of two-level projection-difference schemes. Theorem 2.23. Assume that in the scheme with weights .2.154/ we have a.v, v/ > 0, and the bilinear form b., / is symmetric and positive definite. Then the condition   0.5 is sufficient for the stability of the scheme .2.154/ in Hb .

57

Section 2.5 Stability of projection-difference schemes

Proof. Suppose in the scheme (2.154)

  1 1 y t C .yO C y/, v D  yO C .1   /y D    2 2

then we get

  1 1   b.y t , y t / C .b.y, O y/ O  b.y, y// C a.v, v/ D 0. 2 2 From this equality, under the conditions of the theorem being considered, we obtain the stability of the scheme (2.154) in Hb .

It should be noted that stability of difference schemes with weights can be established in some other more complicated norms.

2.5.6 Stability with respect to the right-hand side As mentioned above, to investigate convergence of the approximate solution to the exact one, a priori estimates with respect to the right-hand side for a discrete problem are of great importance. Here are typical results on stability of projection-difference schemes with respect to the right-hand side. We have investigated above the stability of finite element procedures with respect to the initial data (see the homogeneous scheme (2.137)). Now, in studying two-level projection-difference schemes, we construct typical estimates for stability with respect to the right-hand side. The well-known stability estimates from the theory of difference schemes again serve us as a reference point. We restrict ourselves to simple examples of elementary stability estimates for the approximate solution with respect to the righthand side and the initial data. Let us consider the projection-difference scheme with an inhomogeneous right-hand side:   nC1  yn y , v C a.y n , v/ D .f n , v/, 8v 2 V h , n D 1, 2, : : : . (2.157) b  Again we study the case with a symmetric positive definite form a., / separately. Theorem 2.24. Let in the scheme .2.157/ the bilinear form a., / be symmetric and positive definite. Assume that the condition  (2.158) b.v, v/  ".v, v/ C a.v, v/, 8v 2 V h , n D 1, 2, : : : , 2 holds with an arbitrary positive number ", then the approximate solution satisfies the a priori estimate a.y nC1 , y nC1 /  a.y 0 , y 0 / C

n 1 X  kf k k2 . 2" kD0

(2.159)

58

Chapter 2 Stability of operator-difference schemes

Proof. Write the scheme (2.157) in the form  1 b.y t , v/  a.y t , v/ C a.yO C y, v/ D .f n , v/, 2 2 8v 2 V h ,

n D 1, 2, : : :

and suppose v D 2y t D 2.yO  y/. Taking into account the inequality  2 .f n , y t /  2 "ky t k2 C kf n k2 2" and the condition (2.158), we have  a.y nC1 , y nC1 /  a.y n , y n / C kf n k2 . 2" This inequality implies the required estimate (2.159). Note that this stability estimate with respect to the right-hand side was obtained under slightly stronger assumptions (compare (2.158) with the necessary and sufficient condition (2.140)). Here is an analog of Theorem 2.24 for the inhomogeneous scheme (2.157). The right-hand side will be estimated in the norm of the space that is dual to Ha . This norm is denoted by kvk,a . Theorem 2.25. Let in the scheme .2.157/ the bilinear forms b., / and a., / be symmetric and positive definite. Then, under the condition 1C"  a.v, v/, 8v 2 V h , n D 1, 2, : : : , 2 fulfilled with " > 0, the approximate solution satisfies the estimate b.v, v/ 

a.y nC1 , y nC1 /  a.y0 , y0 / C

n 1C" X  kfk k2,a . 2"

(2.160)

(2.161)

kD0

Proof. Similarly to Lemma 2.4, we have the equality     2 b.y t , y t /  a.y t , y t / C b.y, O y/ O  b.y, y/ 2  C a.yO C y, yO C y/ D 2 .f n , y/. O 2 For the right-hand side of (2.162), we get

(2.162)

O D  .f n , yO C y/ C  2 .f n , y t / 2 .f n , y/    a.yO C y, yO C y/ C kf n k2,a 2 2  3"  C .a.y t , y t / C kf n k2,a . 2 2" In view of (2.160), the substitution of this relation into (2.162) leads to the required a priori estimate (2.161).

59

Section 2.5 Stability of projection-difference schemes

2.5.7 Stability of three-level schemes with respect to the initial data Briefly we shall discuss the issues of stability of three-level projection-difference schemes. Elementary criteria for stability with respect to the initial data will be obtained. The reader can find more general results in special publications. In accordance with a tradition used in the general theory of stability of difference schemes, we start with writing the scheme under consideration in the canonical form and then continue with studying general conditions for stability. For a three-level projection-difference scheme, we have   nC1 y  y n1 , v C r n .y nC1  2y n C y n1 , v/ C an .y n , v/ D .f n , v/, bn 2 8v 2 V h ,

n D 1, 2, : : : ,

(2.163)

where b n ., /, r n ., /, an ., / stand for some real bilinear forms. From (2.163), we seek the approximate solution y n , n D 2, 3, : : : under the condition that y 0 and y 1 are given. Assume that the bilinear forms in (2.163) are constant (independent of n). Consider the three-level projection-difference scheme   nC1  y n1 y , v C r.y nC1  2y n C y n1 , v/ b 2 (2.164) n h C a.y , v/ D 0, 8v 2 V , n D 1, 2, : : : , i.e., first, we will study stability with respect to the initial data. Considering three-level difference schemes, we establish their stability in some complicated enough norms. Let us obtain an a priori estimate for the scheme (2.164) that expresses stability with respect to the initial data. Assume that 1 (2.165) un D .y n C y n1 /, w n D y n  y n1 , 2 then, in view of the identity 1 1 y n D .y nC1 C 2y n C y n1 /  .y nC1  2y n C y n1 /, 4 4 we can rewrite the scheme (2.165) in the form   nC1 w C wn , v C r.w nC1  w n , v/ b 2  nC1  1 u C un (2.166) nC1 n  a.w  w , v/ C a , v D 0, 4 2 8v 2 V h ,

n D 1, 2, : : : .

Selecting in (2.166) v D 2.unC1  un / D w nC1 C w n ,

60

Chapter 2 Stability of operator-difference schemes

we get the equality 1 b.w nC1 C w n , w nC1 C w n / C r.w nC1  w n , w nC1 C w n / 2 (2.167) 1  a.w nC1  w n , w nC1 C w n / C a.unC1 C un , unC1  un / D 0. 4 If the bilinear forms, r., / .r.u, v/ D r.v, u// and a., / are symmetric and the form b., / is non-negative .b.v, v/  0/, then from (2.167) we obtain the inequality E nC1  E n ,

(2.168)

where in view of the notation (2.165) 1 E nC1 D a.y nC1 C y n , y nC1 C y n / 4

1 C r.y nC1  y n , y nC1  y n /  a.y nC1  y n , y nC1  y n /. 4

(2.169)

Under certain restrictions, the quantity E n , which is specified in accordance with (2.169), defines a norm and therefore the inequality (2.168) provides stability of the projection-difference scheme with respect to the initial data. More precisely, we have the following result. Theorem 2.26. Let in the projection-difference scheme .2.163/ the bilinear forms r., / and a., / be symmetric. Then, if the condition 1 b.v, v/  0, a.v, v/ > 0, r.v, v/  a.v, v/ > 0, 8v 2 V h 4

(2.170)

is valid, the a priori estimate 1 1 nC1 C y n k2a C ky nC1  y n k2r  ky nC1  y n k2a ky 4 4 1 n 1 n1 2 n n1 2  ky C y ka C ky  y kr  ky n  y n1 k2a , 4 4

(2.171)

holds, i.e., the projection-difference scheme (2.163) is stable with respect to the initial data. A peculiarity of these three-level schemes is that the norm (see (2.169)) has a complicated structure. In some important cases, it is possible to reduce the class of projection-difference schemes considered here and to employ simpler norms.

2.5.8 Stability with respect to the right-hand side We present some elementary estimates for stability of three-level projection-difference schemes with respect to the initial data and the right-hand side. Instead of (2.164), we

Section 2.5 Stability of projection-difference schemes

61

consider the scheme  nC1  y  y n1 b , v C r.y nC1  2y n C y n1 , v/ C a.y n , v/ D .f n , v/, 2 (2.172) h 8v 2 V , n D 1, 2, : : : . Theorem 2.27. Let in the projection-difference scheme .2.172/ the bilinear forms r., / and a., / be symmetric. Then, under the condition 1 b.v, v/  ".v, v/, a.v, v/ > 0, r.v, v/  a.v, v/ > 0, 8v 2 V h 4

(2.173)

fulfilled with a constant " > 0, the approximate solution satisfies the a priori estimates: E nC1  E 1 C

n 1 X  kf k k2 , 2"

(2.174)

kD1

E nC1

n 1X 1 E C  kf k k2,b , 2

(2.175)

kD1

where k  k,b is the norm of the space that is dual to Hb . Proof. Similarly to the proof of Theorem 2.26 (see (2.166)), we obtain the equality 1 b.w nC1 C w n , w nC1 C w n / C E nC1 D .f n , w nC1 C w n / C E n . 2 To derive the estimate (2.174) with " > 0 and under the condition (2.173), we employ the inequality 1  .f n , w nC1 C w n /  "kw nC1 C w n k2 C kf n k2 . 2 2" As in the proof of (2.175), we utilize 1  .f n , w nC1 C w n /  kw nC1 C w n k2b C kf n k2,b . 2 2 Further arguments are obvious. Further stability estimates for the scheme (2.172) can be obtained under somewhat stronger restrictions on r., /.

2.5.9 Schemes for an equation of first order As an example of the application of the obtained results, we consider the stability conditions for the Cauchy problem for the equation   du (2.176) , v C a.u, v/ D 0, 8v 2 V h , 0 < t  T , c dt

62

Chapter 2 Stability of operator-difference schemes

where c., /, a., / are symmetric, positive definite, and bilinear forms. For (2.176), we use the following three-level scheme:  nC1  u un  un1  un c  C .1   / ,v   (2.177) nC1 n n1 h C a.1 u C .1  1  2 /u C 2 u , v/ D 0, 8v 2 V with weight parameters  , 1 and 2 . The three-level projection-difference scheme (2.177) may be written in the canonical form (2.163) with b.u, v/ D c.u, v/ C .1  2 / a.u, v/,   0.5 1 C 2 r.u, v/ D c.u, v/ C a.u, v/. (2.178)  2 The stability condition (2.170) is valid if and only if 1 1 1  2 ,   , 1 C 2 > . 2 2 The stability condition for the scheme (2.164), (2.178) becomes more evident if we use the inequality (2.179) a.v, v/  c.v, v/, 8v 2 V h with some constant , depending, e.g., on a finite element mesh. The scheme (2.164), (2.178), (2.179) is stable if the following inequality is valid: 1 1 2  1 , 1 C 2 >  .

 2

 A stability condition for schemes with weights is formulated in a similar way for an evolutionary equation of second order (e.g., for the wave equation). 1  2  

Chapter 3

Operator splitting

Various approaches to constructing splitting operators are discussed for typical applied problems. Considering a time-dependent problem for the two-dimensional equation of convection-diffusion, an additive representation of the problem operator is derived employing one-dimensional spatial operators (splitting with respect to spatial variables) or using the separate implementation of operators for diffusion and convection (splitting with respect to physical processes). Domain decomposition methods are created via splitting into subdomains. To solve vector problems, splitting is associated with individual problems for vector components. A triangular decomposition of a problem operator is applied to derive additive schemes for such problems. Some other possibilities to design efficient computational algorithms on the basis of additive schemes are shown on problems of hydrodynamics for a viscous incompressible fluid.

3.1 Time-dependent problems of convection-diffusion Primary problems of the numerical solving of time-dependent problems for the convection-diffusion equation are discussed [63, 100, 153]. In this equation, the convective terms are considered in the divergent, nondivergent, and skew-symmetric forms. Some basic results are presented for a model initial-boundary value problem with Dirichlet boundary conditions for the differential equation of convection-diffusion. These results will serve us as a checkpoint in constructing difference schemes. The investigation of stability and convergence is based on the general theory of stability for operator-difference schemes. Error estimates for difference solutions are obtained in various norms.

3.1.1 Differential problem Time-dependent problems of convection-diffusion are treated as evolutionary operator equations in the corresponding spaces. We start our consideration with a study on properties of differential operators describing the convective and diffusive transport. The focus is on investigating a dependence of the solution of the boundary value problem on time, in particular, on obtaining a priori estimates for the solutions of these time-dependent problems in various norms. As the basic problem, we consider a timedependent problem of convection-diffusion with Dirichlet boundary conditions in a rectangle. The convective terms are written in various forms. A more detailed discussion of these and some other issues can be found in [100, 142, 153].

64

Chapter 3 Operator splitting

We distinguish a class of model time-dependent problems of convection-diffusion with a constant coefficient of diffusive transport (it is independent of time but depends on spatial coordinates). As for coefficients of convective transport, in applied problems they are variable in space and time. Assume that the coefficient at the time derivative equals unity. Many applied problems, e.g., the standard problems of heat transfer, can be reduced to satisfy this assumption. In a rectangle , we consider the time-dependent convection-diffusion equation with the convective terms written in the nondivergent form:   2 2 X @u @ @u @u X v˛ .x, t /  C k.x/ @t @x˛ @x˛ @x˛ (3.1) ˛D1 ˛D1 D f .x, t /,

x 2 ,

t > 0.

This equation is supplemented with homogeneous Dirichlet boundary conditions u.x, t / D 0,

x 2 @,

t > 0.

(3.2)

In addition, the initial data are given: u.x, 0/ D u0 .x/,

x 2 .

(3.3)

The second example is the time-dependent equation of convection-diffusion with the convective transport written in the divergence form:   2 2 X @u X @ @ @u .v˛ .x, t /u/  C k.x/ @t @x˛ @x˛ @x˛ (3.4) ˛D1 ˛D1 D f .x, t /,

x 2 ,

t > 0.

The primary object of our investigation is the convection-diffusion equation with the convective terms written in the skew-symmetric form:  2  @u @ 1X @u C .v˛ .x, t /u/ C v˛ .x, t / @t 2 ˛D1 @x˛ @x˛ (3.5)   2 X @ @u  k.x/ D f .x, t /, x 2 , t > 0. @x˛ @x˛ ˛D1 We consider a set of functions u.x, t / that satisfy the boundary condition (3.2). The time-dependent convection-diffusion problem may be formulated as the operatordifferential equation du C Au D f .t /, A D A.t / D C .t / C D, t > 0 (3.6) dt using the following notation for the operators of diffusive and convective transport.

Section 3.1 Time-dependent problems of convection-diffusion

65

For the diffusive transport operator, we use the expression   2 X @ @u Du D  k.x/ . @x˛ @x˛ ˛D1 For the convective transport operators C D Cˇ , ˇ D 0, 1, 2, suppose:  2  1X @u @ C .v˛ .x, t /u/ , C0 u D v˛ .x, t / 2 ˛D1 @x˛ @x˛ C1 u D C2 u D

2 X ˛D1 2 X

v˛ .x, t /

@u , @x˛

@ .v˛ .x, t /u/. @x˛ ˛D1

For the evolutionary equation (3.6), we consider the Cauchy problem u.0/ D u0 .

(3.7)

We recall some essential properties of the differential operators of diffusive and convective transport. In H D L2 ./, for the diffusive transport operator, we have D D D   0 E.

(3.8)

For the operators of convective transport, we need the estimates: j.C u, u/j  M1 kuk2 ,

(3.9)

kC uk2  M2 .Du, u/

(3.10)

with the corresponding constants. For example, for the operator of convective transport in the skew-symmetric form C D C0 ), we have in the inequality (3.9) the constant M1 D 0, i.e., the operator is skew-symmetric. We restrict ourselves to elementary a priori estimates for the time-dependent problem (3.6), (3.7), for which we will construct discrete analogs. Theorem 3.1. For the problem .3.6/, .3.7/ under the conditions .3.8/–.3.10/, we have the a priori estimates: Z 1 t 2 0 2 exp.2M1 .t   //kf . /k2D 1 d , (3.11) ku.t /k  exp.2M1 t /ku k C 2 0   Z 1  t 1 ku.t /k  exp M2 t ku0 k C exp (3.12) M2 .t   / kf . /kd , 4 4 0 Z N 1 t kru.t /k2  exp.M2 t /kru0 k2 C exp.M2 .t   //kf . /k2 d , (3.13) 0 where  k.x/  , N x 2  [ @.

66

Chapter 3 Operator splitting

Proof. Multiplying equation (3.6) scalarly by u.t /, we get 1d kuk2 C .Du, u/ D .C u, u/ C .f , u/. 2 dt Taking into account (3.9) and the inequality 1 .f , u/  .Du, u/ C kf k2D 1 , 4 from (3.14) it follows that

(3.14)

1 d kuk2  2M1 kuk2 C kf k2D 1 . dt 2 Using Gronwall’s lemma, we obtain from this inequality the required estimate (3.11). From (3.10), we have 1 j  .C u, u/j  kC ukkuk  .Du, u/ C M2 kuk2 . 4 This allows to obtain from (3.14) the inequality 1 d kuk  M2 kuk C kf k, dt 4 which immediately implies the estimate (3.12). It remains to derive (3.13). To do this, multiply equation (3.6) scalarly by du=dt and obtain  2      du    C 1 d .Du, u/ D  C u, du C f , du .  dt  2 dt dt dt For the right-hand side, we have     2   du  1 1 du du 2 2  C f,   C u,  dt  C 2 kC uk C 2 kf k . dt dt In view of (3.10), we get the inequality d .Du, u/  M2 .Du, u/ C kf k2 . dt

(3.15)

By 2 N , kruk2  .Du, u/  kruk

from (3.15), we obtain the desired estimate (3.13). The resulting estimates (3.11)–(3.13) provide the continuity of the solution of (3.6), (3.7) with respect to the initial data and the right-hand side. In these estimates, the essential result is that for the solution norm of the problem with the homogeneous righthand side, an exponential growth is allowed with a growth increment that depends on the constants M1 , M2 . It is necessary to allow such a behavior for the solution at

Section 3.1 Time-dependent problems of convection-diffusion

67

the discrete level. Thus, we need to introduce the concept of %-stability for difference schemes. Considering boundary value problems for both parabolic equations of the second order in space and second-order elliptic equations, special attention is paid to the maximum principle. The corresponding statement for the time-dependent case considered here is formulated as follows. Theorem 3.2. Assume that in the Cauchy problem .3.6/, .3.7/, the right-hand side f .x, t / > 0 .f .x, t / < 0/ and the initial data u0 .x/  0 .u0 .x/  0/, then u.x, 0/  0 .u.x, 0/  0/ for all x 2  and t > 0. The proof is completely analogous to proving the maximum principle for secondorder elliptic equations. Note also that it is possible to use the maximum principle in a stronger form that employs weak inequalities for the right-hand side, i.e., the nonnegativeness of the solution takes place under the condition of the non-negativity of the right-hand side and the initial data. Here are some a priori estimates for the convection-diffusion problem (3.6), (3.7), which are derived from the maximum principle [37, 80, 106]. In the time-dependent problems with Dirichlet boundary conditions, we can easily construct a majorant function. The solution of the time-dependent convection-diffusion problem (3.6), (3.7) satisfies the following a priori estimate in L1 /: Z ku.x, t /k1  ku.x, 0/k1 C

t

0

kf .x,  /k1 d .

(3.16)

We also give an estimate for the convection-diffusion equation with the convective terms in the divergent form, i.e., the estimate in L1 /. Theorem 3.3. The solution of the problem .3.2/–.3.3/ satisfies the a priori estimate Z ku.x, t /k1  ku.x, 0/k1 C

t

kf .x,  /k1 d .

(3.17)

0

Proof. Denote by u.x, N t / the solution of the convection-diffusion problem with the N t / is a majorant funcright-hand side jf .x, t /j and the initial data ju0 .x/j. Then u.x, tion for the original problem. Further, we integrate the equation for u.x, N t / over the whole domain . This concludes the proof. The estimates (3.16), (3.17) complement the above a priori estimates (3.11)–(3.13) in the Hilbert spaces L2 ./ and W21 ./.

68

Chapter 3 Operator splitting

3.1.2 Semi-discrete problem To solve numerically the time-dependent convection-diffusion problem in the rectangle , we use a uniform rectangular grid ¯ ® !N ˛ D x˛ j x˛ D i˛ h˛ , i˛ D 0, 1, : : : , N˛ , N˛ h˛ D l˛ , ¯ ® !˛ D x˛ j x˛ D i˛ h˛ , i˛ D 1, 2, : : : , N˛  1, N˛ h˛ D l˛ . On the set of grid functions that are equal to zero on @! .!N D ! [ @!/, we define the Hilbert space H D L2 .!/ with the scalar product and norm X .y, w/  y.x/w.x/h1 h2 , kyk  .y, y/1=2 . x2!

Assume that the convection-diffusion problem has a smooth enough coefficient k.x/ and smooth solutions, then we can approximate the differential operator D with an error of O.jhj2 / as follows: DD

2 X

D .˛/ ,

D .˛/ y D .a.˛/ yxN ˛ /x˛ , ˛ D 1, 2, x 2 !,

(3.18)

˛D1

where, e.g., a.1/ .x/ D k.x1  0.5h1 , x2 /,

a.2/ .x/ D k.x1 , x2  0.5h2 /.

For the difference operator of diffusive transport, we have D D D   ıE, where ı D 8

2 X 1 , l2 ˛D1 ˛

D  M4 E,

M4 D 4 N

(3.19)

2 X 1 . h2 ˛D1 ˛

The convective terms in the convection-diffusion equation are approximated with the second order in space on the basis of central difference derivatives and staggered spatial grids to define the velocity components. The advantages of such approximations are discussed in detail, e.g., in [153]. For the nondivergent operator of convective transport, we set C1 D

2 X

.˛/

C1 ,

˛D1

where

 1  .1/ b .x1  0.5h1 , x2 /yxN 1 C b .1/ .x1 C 0.5h1 , x2 /yx1 , 2   1 .2/ C1 y D b .2/ .x1 , x2  0.5h2 /yxN 2 C b .2/ .x1 , x2 C 0.5h2 /yx2 . 2 .1/

C1 y D

(3.20)

Section 3.1 Time-dependent problems of convection-diffusion

69

In the simplest case, where the solution and velocity components of the differential problem are appropriately smooth, we can assume, e.g., b˛ .x/ D v˛ .x/, ˛ D 1, 2. For the convection-diffusion equation with the divergent convective terms, we employ the discrete operator 2 X .˛/ C2 , (3.21) C2 D ˛D1

where now .1/

1 .1/ b .x1 C 0.5h1 , x2 /.y.x1 C h1 , x2 / C y.x1 , x2 // 2h1 1 .1/ b .x1  0.5h1 , x2 /.y.x1  h1 , x2 / C y.x1 , x2 //,  2h1

.2/

1 .2/ b .x1 , x2 C 0.5h2 /.y.x1 , x2 C h2 / C y.x1 , x2 // 2h2 1 .1/ b .x1 , x2  0.5h2 /.y.x1 , x2  h2 / C y.x1 , x2 //.  2h2

C2 y D

C2 y D

For the operator

1 C0 D .C1 C C2 /, 2

we obtain the expression C0 D

2 X

.˛/

C0 ,

(3.22)

˛D1

where .1/

C0 y D

1 .1/ b .x1 C 0.5h1 , x2 /y.x1 C h1 , x2 / 2h1 1 .1/ b .x1  0.5h1 , x2 /y.x1  h1 , x2 /,  2h1

1 .2/ b .x1 , x2 C 0.5h2 /y.x1 , x2 C h2 / 2h2 1 .1/  b .x1 , x2  0.5h2 /y.x1 , x2  h2 /. 2h2 It follows immediately that, as in the continuous case, we have .2/

C0 y D

C1 D C2 ,

C0 D C0 .

For the inequality (3.9), we have the discrete analog

where

j.Cy, y/j  M1 kyk2 ,

(3.23)

8 ˇ D 0, < 0, Mˇ D 1 : max jdivh bj, ˇ D 1, 2. 2 x2!

(3.24)

70

Chapter 3 Operator splitting

Here the discrete operator of divergence is defined as follows: divh b D

2 X

divh b˛ ,

˛D1

where

1 .1/ .b .x1 C 0.5h1 , x2 /  b .1/ .x1  0.5h1 , x2 //, h1 1 divh b2 D .b .2/ .x1 , x2 C 0.5h2 /  b .2/ .x1 , x2  0.5h2 //. h2 There is also a discrete analog of the subordination inequality (3.10), i.e., divh b1 D

kCyk2  M2 .Dy, y/

(3.25)

with a constant M2 that is consistent with the constant M2 from the differential problem. For instance, for the nondivergent operator of convective transport, we have ² ³ 2 .1/ 2 .2/ 2 M2 D max max.b .x1  0.5h1 , x2 // , max.b .x1 , x2  0.5h2 // . x2!N x2!N The subordination constant M2 demonstrates the fundamental property – it is independent of a computational grid. It depends on both the value of the coefficients of convective transport v˛ .x/, ˛ D 1, 2 (the velocity) and div v, more precisely, it depends on their difference approximations. Using the above discretization in space, we obtain the operator-differential equation dw C Aw D .t /, dt

A D A.t / D C.t / C D,

0 < t  T,

(3.26)

defined on the set of grid functions w.t / 2 H . The initial data have the form w.t / D u0 .

(3.27)

The problem (3.26), (3.27) is a semi-discrete analog of the problem (3.6), (3.7).

3.1.3 Two-level schemes To solve numerically the problem (3.26), (3.27), we consider the two-level scheme with weights y nC1  y n C C.1 y nC1 C .1  1 /y n /  C D.2 y nC1 C .1  2 /y n / D ' n , y0 D u0 .

(3.28) t n 2 ! , (3.29)

71

Section 3.1 Time-dependent problems of convection-diffusion

Here, e.g., we have C D C.0.5.t nC1 C t n //,

' n D .0.5.t nC1 C t n //.

Among the most important variants of the difference scheme with weights (3.28), (3.29), we highlight the scheme with equal weights (1 D 2 ) and the scheme where convective transport is taken from a previous time level (1 D 0). We start with the convective transport operator in the skew-symmetric form, i.e., C D C  D C0 . Problems with the convective transport operator in the nondivergent (C D C1 ) and divergent (C D C2 ) forms will be considered later. Assume that in the difference scheme (3.28), we have 1 D 2 D  .

(3.30)

First, we will derive criteria for stability of difference schemes for time-dependent convection-diffusion problems with respect to the initial data only. A priori estimates for stability with respect to the right-hand side are important to investigate convergence of difference schemes. Thus, taking into account (3.30), instead of (3.28), we consider the difference scheme y nC1  y n C .C0 C D/.y nC1 C .1   /y n / D 0, 

t n 2 ! .

(3.31)

The scheme (3.29), (3.31) may be written as a homogeneous (with respect to the right-hand side) difference scheme in the canonical form: B

y nC1  y n C Ay n D 0, 

t n 2 !

(3.32)

A D C0 C D > 0.

(3.33)

with operators B D E C  A,

The main peculiarity of difference schemes for the convection-diffusion equation is connected with non-self-adjointness of the operators B and A. Therefore, it is impossible to use the above results on stability of operator-difference schemes, which were formulated for constant self-adjoint operators. The second important feature is associated with the fact that operators of the difference scheme are variable in time. We consider the problems with the time-dependent difference operator of convective transport. To obtain a priori estimates for such problems, it is often necessary to additionally require Lipschitz continuity of the difference operators with respect to time. Suppose, e.g., that in the scheme (3.32), (3.33), the operator of convective transport is constant. Using the fundamental theorem on stability of two-level difference schemes, it is possible to formulate a criterion for stability in HA A .

72

Chapter 3 Operator splitting

Theorem 3.4. The scheme with weights .3.32/, .3.33/ with A 6D A.t / is stable in HA A if and only if the following necessary and sufficient condition is satisfied:   1 .Dy, y/ C   (3.34)  k.C0 C D/yk2  0. 2 Proof. To use general results of the stability theory for operator-difference schemes, it is necessary to go to the problem y BQ

 yn Q n D 0, C Ay 

nC1

t n 2 ! ,

(3.35)

where AQ is a constant self-adjoint operator. To reduce the scheme (3.32), (3.33) with the constant convective transport operator C0 , multiply (3.32) by A . This gives the difference scheme (3.35) with BQ D A C  A A,

AQ D A A.

The necessary and sufficient condition for stability in HAQ , i.e.,  BQ  AQ 2 leads to the inequality (3.34). In view of (3.34), the scheme (3.32), (3.33) is unconditionally stable under the constraint   0.5. Now we will formulate conditions for stability of the schemes with 0   < 0.5 in the form of restrictions on a permissible time step. For the second term in the left-hand side of (3.34), we employ the estimate k.C0 C D/yk2  2.kC0 k2 C kDyk2 /. By (3.19), we have kDyk2  M3 .Dy, y/ that in combination with (3.25) gives k.C0 C D/yk2  2.M2 C M3 /.Dy, y/. The condition (3.34) is satisfied with   0 D

1 . .1  2 /.M2 C M3 /

Taking into account M2 D O.1/, M3 D O.jhj2 /, we obtain that the maximal step  0 D O.jhj2 /. If (3.34) is fulfilled, then the following estimate for stability in HA A holds: kAy nC1 k  kAy n k.

Section 3.1 Time-dependent problems of convection-diffusion

73

In the case of variable operators, it seems reasonable to consider %-stability assuming the Lipschitz continuity of A.t /, i.e., supposing the fulfillment of the estimate k.A.t C  /  A.t //yk  m kA.t /yk with a positive constant m. However, for convection-diffusion problems with variable operators, it is best to orient to estimates for stability in H , HD (see the corresponding estimates for the solution of the differential problem). Theorem 3.5. The scheme with weights .3.32/, .3.33/ is stable in H if and only if the necessary and sufficient operator inequality .3.34/ is satisfied. Proof. In view of A > 0, there exists A1 . Multiplying (3.34) by A1 , we get the scheme (3.35), where BQ D A1 C  E, AQ D E. The necessary and sufficient condition for stability in H D HAQ has the form of the inequality   1 A1 C   E  0. 2 Multiplying it from the left by A and from the right by A (in doing so, this inequality remains valid), we obtain   1  A A  0. A C  2 For (3.33), this inequality is just the required inequality (3.34). We now consider the case where the skew-symmetry of the difference operator of convective transport is not valid. We will study the problem with the convective transport written in the nondivergent form, i.e., C D C1 . The case where the convective transport in the divergence form (C D C2 ) is investigated in a similar way. Let us examine the scheme (3.32), where B D E C  A,

A D C1 C D.

(3.36)

It is important to distinguish two classes of problems. The simplest case is associated with the assumption that the operator A is non-negative. Such a situation takes place, e.g., if M1 M0   0 – convective transport has only an insignificant effect. Another case deals with slightly compressible flows, where A  0 under the condition M2 M0   0. In this situation, we can apply the results of the unconditional stability for the difference scheme (3.32), (3.36) in H with the constraint   0.5. Theorem 3.6. The scheme with weights .3.32/, .3.36/ with A  0 is stable in H if and only if   0.5.

74

Chapter 3 Operator splitting

Proof. To prove this statement, we rewrite the scheme (3.32), (3.36) in the form y nC1  y n C Av nC1 D 0,  where

n D 0, 1, : : : ,

(3.37)

  1 1 v D y C .1  y / D   y t C .y nC1 C y n /, 2 2 y nC1  y n yt D .  nC1

nC1

n

Scalar multiplication of equation (3.37) by v nC1 yields   1 1  .y t , y t / C A.v nC1 , v nC1 / C ..y nC1 , y nC1 /  .y n , y n // D 0.  2 2 From this inequality, under the condition of the theorem, the estimate follows: ky nC1 k  ky n k, i.e., the scheme (3.32), (3.36) is stable in H . The solvability of the scheme (3.32), (3.36) (B > 0), in view of a possible negativeness of the operator A, takes place under the constraint of an appropriately small time step. Taking into account (3.19), (3.23) with M1 M0  > 0, we get the following insignificant restriction on a time step:   1 D

M0 .  .M1 M0  /

(3.38)

In this case (see the differential problem), it is necessary to be oriented to obtaining an appropriate estimate that expresses conditions for %-stability. Earlier we formulated the necessary and sufficient condition for %-stability in the case with the constant self-adjoint operators B and A. Therefore, our study will be based on the schemes with weights of type (3.32), (3.33) considered above. Let us define new grid functions v n : y n D %n v n ,

n D 0, 1, : : : ,

% > 0.

(3.39)

A condition for %-stability for y n is evidently equivalent to stability (% D 1) for v n . Substitution of (3.39) into (3.32) yields the difference scheme v BQ where

 vn Q n D 0, C Av 

nC1

BQ D %E C %A,

t n 2 ! ,

%1 AQ D E C .1 C  .%  1//A. 

(3.40) (3.41)

Section 3.1 Time-dependent problems of convection-diffusion

75

It is possible to use the following representation for the operators of the difference scheme (3.40): Q BQ D G C Q  A, (3.42) treating the scheme (3.40) as a scheme with weights. In view of the representation (3.41), we obtain in (3.42): % % E, Q D . (3.43) GD 1 C  .%  1/ 1 C  .%  1/ Similarly to Theorem 3.6, we prove the stability of the scheme (3.40), (3.42) in HG , i.e., in H with Q  0.5 under the constraint AQ  0. Taking into account (3.43), we get the desired condition on a weight of the difference scheme (3.40), (3.41): 1 . (3.44)  1C% The non-negativity of the operator AQ is connected with an appropriate choice of %. In view of the stability estimate for the differential problem (see Theorem 3.1), it is natural to set (3.45) % D 1 C M1  . Taking into account the estimate A

M1 M0  E, M0

the condition AQ  0 (see (3.41)) will be valid under the constraint M1 M0  .1 C  M1 /.M1 M0  /  0. This inequality yields the following restriction on a permissible time step: .   2 D M1 .M1 M0  /

(3.46)

A comparison with the estimate (3.38) shows that the time step restriction (3.46) is slightly stronger (2 < 1 as, we recall, M1 M0 > ). Summarizing, we obtain the following statement. Theorem 3.7. The scheme with weights .3.32/, .3.34/ with the constraint M1 M0 > is %-stable in H , where % is defined according to .3.45/, if a weight  satisfies the restriction .3.44/ and a time step meets the condition .3.46/. This statement complements Theorem 3.6, which ensures the stability of the scheme (3.32), (3.36) under the constraint M1 M0  in H with   0.5. Possible nonnegativity of the operator A D C1 C D leads to the situation where we must use %-stability. In addition, we impose weak restrictions on a time step, not associated with a discretization in space (see (3.46)).

76

Chapter 3 Operator splitting

In solving convection-diffusion problems, it is natural to focus on difference schemes in which the most inconvenient part of the operator A (it is, of course, the convective transport operator) is taken from the previous time level. Such explicitimplicit schemes from the above class of two-level schemes with weights are considered here [7, 113]. Suppose now that in the difference scheme (3.28), we have 1 D 0,

2 D  .

(3.47)

The homogeneous (' n D 0) scheme (3.28), (3.47) is reduced to the canonical form (3.32) if B D E C  D, A D C C D. (3.48) For any  > 0, we have B > 0, and therefore the discrete equation (3.32) is solvable at every time level. Let us formulate a sufficient condition for %-stability of the difference scheme for the convection-diffusion equation in HD [210]. Theorem 3.8. The solution of the explicit-implicit scheme .3.32/, .3.46/ with   0.5 satisfies the estimate (3.49) ky nC1 kD  %ky n kD , where

M2 , 4 and M2 is the constant from the inequality .3.25/. %D1C

(3.50)

Proof. Multiply (3.32) scalarly by 2y t D 2.y nC1 y n / and, in view of (3.48), obtain the energy identity  ..2B  D/y t , y t / C .Dy nC1 , y nC1 /  .Dy n , y n / C 2 .Cy n , y t / D 0. (3.51) Taking into account the representation (3.48) and the constraint   0.5, from (3.51) the inequality follows: 2 .y t , y t / C .Dy nC1 , y nC1 /  .Dy n , y n /  2 j  .Cy n , y t /j. In view of (3.25), the right-hand side is evaluated as follows: 1 M2 .Dy n , y n /. j  .Cy n , y t /j  ky t k2 C kCy n k2  ky t k2 C 4 4 Substitution into (3.52) yields   M2 nC1 nC1 .Dy ,y / 1C  .Dy n , y n /. 2 Therefore, in view of inequality   M2 M2 1=2 1C  , 1C 2 4 we obtain the desired stability estimate (3.49), (3.50).

(3.52)

77

Section 3.2 Splitting operators in convection-diffusion problems

The %-stability estimate (3.49), (3.50), derived here, is fully consistent with the corresponding estimate for the differential problem. An important point is that, in contrast to Theorem 3.6, we obtained stability with the standard restrictions on a weight  in a stronger norm. Moreover, the implementation of the explicit-implicit scheme is much simpler from the computational point of view – we must invert a self-adjoint elliptic grid operator. Another class of unconditionally stable explicit-implicit schemes for convection-diffusion problems are discussed in the article [212].

3.2 Splitting operators in convection-diffusion problems Standard approaches to construct splitting operators are discussed using a model problem of convection-diffusion as an example. First, possibilities of splitting with respect to spatial variables are studied for the problem in a rectangle. The second approach deals with the reduction to simpler problems via splitting of a problem operator into operators of diffusion and convection. In this case, we speak of splitting with respect to physical processes.

3.2.1 Splitting with respect to spatial variables A discretization in space of the above-considered model convection-diffusion problems in a rectangle yields the following Cauchy problem for the operator-differential equation of first order (see (3.26), (3.27)): du C Au D f .t /, dt

0 < t  T,

u.t / D u0 .

(3.53) (3.54)

The problem operator A is represented as A D A.t / D C.t / C D,

(3.55)

where D denotes a discrete operator of diffusive transport, whereas C stands for a discrete operator of convective transport. Additive schemes under consideration are implicitly or explicitly associated with an additive representation of the operator A: AD

p X

A˛ .

(3.56)

˛D1

The nature of such splitting may be different. Let us start with the classical case of splitting with respect to spatial variables. In view of (3.18) and (3.20)–(3.22), for individual terms in (3.56) with p D 2, we put (3.57) A˛ D D .˛/ C C .˛/ , ˛ D 1, 2.

78

Chapter 3 Operator splitting

The operators A˛ , ˛ D 1, 2 are one-dimensional; they are associated with solving a set of boundary value problems for a one-dimensional parabolic equation (a onedimensional convection-diffusion equation) at every time level. Standard operator-splitting schemes are based on the assumption that the operators A˛ , ˛ D 1, 2, : : : , p are non-negative. For the model convection-diffusion problems investigated here, this constraint is unconditionally valid for the operator of convective transport in the skew-symmetric form C D C0 . In this case, in (3.56), (3.57), we have .˛/

A˛ D D .˛/ C C0 , and also .˛/

C0

˛ D 1, 2,

D .C0 / . .˛/

For the operators of convective transport in the nondivergent (C D C1 ) and divergent (C D C2 ) forms (see (3.23), (3.24)), the unconditional non-negativity of the operator (especially for its individual terms) is not guaranteed. In this case, the use of standard implicit schemes leads to restrictions on a time step. To construct unconditionally stable schemes, we need to modify standard approximations in time. Some possibilities in this direction are discussed below.

3.2.2 Splitting with respect to physical processes To develop additive schemes for the Cauchy problem (3.53), (3.54), we start directly with the splitting (3.55). We use the representation (3.56) with p D 2, where A1 D D,

A2 D C .

(3.58)

In this case, the individual operators are associated with physical processes of a different nature, i.e., with the diffusive and convective transport. Therefore, in the case (3.58), we speak of splitting with respect to physical processes. If the discrete operator of convective transport is written in the skew-symmetric form (the boundary value problem (3.2), (3.3), (3.5)), then C D C0 ,

C0 D C0 .

(3.59)

In the case (3.58), (3.59), we come to the standard splitting (3.56), where A˛  0, ˛ D 1, 2. In solving the convection-diffusion problem with the convective transport in the nondivergent form (C D C1 – the problem (3.1)–(3.3)), we have

where (see (3.20)–(3.24))

C D C0 C R,

(3.60)

1 Ry D  divh b y. 2

(3.61)

Section 3.2 Splitting operators in convection-diffusion problems

79

For the convection-diffusion problem with the convective transport in the divergent form (C D C2 – the problem (3.2)–(3.4)), the representation (3.60) looks like this: Ry D

1 divh b y. 2

(3.62)

The operator R may be treated as an effective reaction coefficient. Thus, we obtain the problem of convection-diffusion-reaction (3.53), (3.54), where A D D C C0 C R.

(3.63)

In constructing approximations in time for the problem (3.53), (3.54), (3.63), we emphasize that the discrete operator of reaction is semibounded from below: R  E, where

(3.64)

1 D  min divh b 2 x2!

in the case of (3.61) and

1 max divh b 2 x2!

D

for (3.62). In view of (3.63), we can apply various two- and three-component splittings of (3.56). For instance, for p D 2, we can put A1 D D,

A2 D C0 C R.

For p D 3, it is possible to treat the effect of the compressibility of a medium (reaction) as a separate stage: A1 D D C R, A1 D D,

A2 D C 0 .

A2 D C 0 ,

(3.65)

A3 D R.

Some difficulties arise for the problems (3.63), (3.64) with < 0. In this case, we cannot guarantee the non-negativity of the operator A and, moreover, this is true for the terms A˛ , ˛ D 1, 2, : : : , p in the representation (3.56). The simplest way to calculate the stage of reaction is to use the three-component splitting (3.65), where the solution of the corresponding subproblem for the equation dy C Ry D 0, dt

t n < t  t nC1 ,

in view of (3.61), (3.62), can be obtained, e.g., in an analytical way. Some modifications of standard implicit schemes can be used for this kind of problem, too.

80

Chapter 3 Operator splitting

3.2.3 Schemes for problems with an operator semibounded from below Let us consider in a finite-dimensional Hilbert space H the Cauchy problem for the operator-differential first-order equation du C Au D f .t /, dt

t > 0,

(3.66)

u.0/ D u0 ,

(3.67)

where a constant operator A is semibounded from below: A  ıE.

(3.68)

The sign of ı may be arbitrary. The solution of the problem (3.66)–(3.68) satisfies the following estimate for stability with respect to the right-hand side and the initial data:   Z t exp.ı /kf . /kd . (3.69) ku.t /k  exp.ıt / ku0 k C 0

Using the standard two-level scheme with weights y nC1  y n C A.y nC1 C .1   /y n / D ' n ,  y 0 D u0 ,

t n 2 ! ,

(3.70) (3.71)

we focus on the corresponding conditions for -stability. Moreover, the numerical implementation of the scheme (3.70), (3.71) is connected with the inversion of the discrete operator B D E C  A. For the existence of the discrete operator B 1 , a necessary and sufficient condition has the form B > 0. By (3.68), we get the restriction on a time step: 1 , ı 0,

(3.74) (3.75)

81

Section 3.2 Splitting operators in convection-diffusion problems

where fQ.t / D exp.ıt /f .t /. The corresponding stability estimate for (3.74), (3.75) has (see (3.69)) the form Z t kfQ. /kd . (3.76) ku.t Q /k  ku0 k C 0

In the problem (3.68), (3.74), (3.75), we have AQ  0, and therefore usual schemes with weights can be applied to solve it numerically. Investigating the class of two-level operator-difference schemes, we put yQ nC1  yQ n Q yQ nC1 C .1   /yQ n / D 'Q n , C A. 

t n 2 ! ,

yQ 0 D u0 , where, e.g.,

(3.77) (3.78)

'Q n D fQ. t nC1 C .1   /t n /.

The scheme (3.77), (3.78) is unconditionally stable in H under the constraint   0.5, and the difference solution satisfies the a priori estimate (see (3.76)) in the form kyQ nC1 k  kyQ n k C  k'Q n k.

(3.79)

Let us study again the difference solution of the problem (3.66)–(3.68). In view of (3.73), we put y n D exp.ıt n /yQ n . The substitution into (3.77), (3.78) yields the difference scheme y nC1  exp.ı /y n C .A  ıE/.y nC1 C .1   / exp.ı /y n /  D exp..  1/ı /' n , t n 2 ! ,

(3.80)

y 0 D u0 .

(3.81)

The main result may be formulated as the following statement. Theorem 3.9. The scheme .3.68/, .3.80/, .3.81/ with   0.5 is unconditionally -stable in H with  D exp.ıt n  /, and the solution satisfies the estimate ky nC1 k  ky n k C  exp..  1/ı /k' n k.

(3.82)

Proof. The estimate (3.82), which is consistent with the estimate (3.69) for the differential problem, follows immediately from the estimate (3.79).

82

Chapter 3 Operator splitting

3.3 Domain decomposition methods Domain decomposition methods are essential in numerically solving boundary value problems for PDEs on parallel computing systems. Iteration-free schemes of domain decomposition are in common use to take into account peculiarities of time-dependent problems. Regionally additive schemes are constructed using various classes of splitting schemes. We emphasize a class of domain decomposition methods that is based on a partition of unity used to divide a domain into subdomains. The investigation is conducted considering as an example the Cauchy problem for evolutionary equations of the first and second order with a non-negative self-adjoint operator in a finite Hilbert space. Problems with a non-self-adjoint operator – convection-diffusion problems – are discussed separately.

3.3.1 Preliminaries The theory and practice of iterative solving of steady-state boundary value problems for PDEs are comprehensively presented in [96, 108, 165, 184]. Various variants of subdomains – either overlapping or non-overlapping – are discussed in these books. As for time-dependent problems, we can employ standard implicit approximations in time and solve the corresponding grid problems at a new time level via domain decomposition methods for steady-state problems. Taking into account specific features of time-dependent problems (see, e.g., an implementation on the basis of Schwarz methods [19,20]), we obtain optimal iterative methods of domain decomposition where the number of iterations does not depend on steps of discretization in time and space. Iteration-free domain decomposition methods explore peculiarities of time-dependent problems in the most efficient way. In some cases, it is possible [77, 78], without loss of accuracy of the approximate solution, to do only one iteration of the alternating Schwarz method in solving boundary value problems for a parabolic second-order equation. We can show that iteration-free schemes of domain decomposition are associated with various variants of additive schemes (splitting schemes) – see regionally additive schemes in [136]. Domain decomposition methods for solving time-dependent problems can be classified by (i) the method of domain decomposition used, (ii) the choice of decomposition operators (exchange of boundary conditions), and (iii) the splitting scheme employed. For differential problems, it is natural to choose domain decomposition methods D

p [

˛ ,

˛ D ˛ [ @˛ ,

˛ D 1, 2, : : : , p

(3.83)

˛D1

with overlapping subdomains (˛ˇ  ˛ \ ˇ ¤ ¿) and with no overlap (˛ˇ D ¿) [108, 184]. Non-overlapping subdomain methods are associated with an explicit formulation of data exchanges on interfaces. These methods are useful in solving prob-

83

Section 3.3 Domain decomposition methods

lems where each separate subdomain has its own computational grid (triangulation). To construct homogeneous numerical algorithms, overlapping subdomain methods are preferable. In the case of the minimal overlap, where the width of an overlapping region is equal to the grid step (˛ˇ D O.h/), overlapping subdomain methods may be treated as non-overlapping subdomain methods with the appropriate data exchange on the interfaces. The domain decomposition (3.83) is associated with the corresponding additive representation of the problem operator AD

p X

A˛ .

(3.84)

˛D1

In this case, the operator A˛ is connected with solving some problem in the subdomains ˛ , ˛ D 1, 2, : : : , p. The most common approach to create decomposition operators for solving boundary value problems for PDEs is based on a partition of unity for the computational domain employed. For the decomposition (3.83), we can associate with each individual subdomain ˛ the function ˛ .x/, ˛ D 1, 2, : : : , p such that ² > 0, x 2 ˛ , ˛ .x/ D ˛ D 1, 2, : : : , p, (3.85) 0, x … ˛ , and also p X ˛ .x/ D 1, x 2 . (3.86) ˛D1

Let, e.g., the problem operator A be the diffusion operator: A D  div k.x/ grad ,

x 2 .

(3.87)

Then we can specify the operators of decomposition by one of the following three basic forms: A˛ D ˛ A,

(3.88)

A˛ D  div k.x/˛ .x/ grad ,

(3.89)

A˛ D A ˛ ,

(3.90)

˛ D 1, 2, : : : , p.

This technique was introduced in the works [81] (the decomposition (3.89)) and [186] (the decomposition (3.88)–(3.90)); the results of more recent studies are summarized in the books [136,151]. Various variants of decomposition operators differ by distinct types of data exchanges on interfaces. They ensure the convergence of the approximate solution in various spaces of grid functions. The construction of decomposition operators for time-dependent problems with non-self-adjoint operators requires special attention [148, 152, 191]. To solve time-dependent problems with the splitting (3.84), various operatorsplitting schemes are used. In the theory of additive operator-difference schemes

84

Chapter 3 Operator splitting

[94, 131, 151, 218], we emphasize the simplest case of two-component splitting. In this case, we construct unconditionally stable factorized splitting schemes, such as the classical ADI methods, predictor-corrector schemes etc. Two-component regionally additive schemes are constructed and studied in [81, 186, 189] as well as in the abovementioned works [148, 152, 191] for convection-diffusion problems. In computational practice, including the application of domain decomposition methods, the splitting of the problem operator into the sum of three or more noncommutative operators (p > 2 in (3.84)) is of great interest. The classical schemes [94,131,218] of multicomponent splitting are based on the concept of summarized approximation. Additively averaged schemes of summarized approximation [50,151] are more explicitly oriented to parallel computing. Regionally additive schemes of componentwise splitting are studied in [213]. A variant of two-component splitting with the Crank– Nicolson scheme for individual subproblems with the minimal overlap and decomposition (3.89) is investigated in the paper [29]. Nowadays, schemes of full approximation are in common use for general multicomponent splitting. In this regard, we highlight the regularized additive schemes [150], where a stability condition is achieved by some perturbation of operators of a difference scheme. In vector additive schemes [1, 193], instead of one equation, we solve a system of similar equations. Such schemes are also constructed for the evolutionary equations of second order [3,141]; vector regionally additive schemes are investigated in the works [143, 214]. Special attention should be given to the paper [196], where more general regularized schemes of domain decomposition are examined including various constructions both for the splitting operators and for the operators of the discrete problem at a new time level. Among the other methods of domain decomposition for solving boundary value problems for parabolic equations, we note explicit-implicit methods, which are considered with slight variations in many papers (see, e.g., [67–69, 174, 220, 221]). In this case, domain decomposition is conducted without overlap and the transition to a new time level is implemented as follows. First, the approximate solution on the common boundaries of subdomains is predicted using the explicit scheme. Next, these boundary conditions are used to evaluate the approximate solution inside the individual subdomains. And finally, implicit schemes are employed at the final stage to correct data on interfaces. As will be shown below, such domain decomposition methods completely fit in the above-mentioned general technique with the choice of operators according to the decomposition (3.88). Here we restrict ourselves to the construction of decomposition operators for parabolic and hyperbolic equations with a self-adjoint elliptic operator of second order and for parabolic equations with a non-self-adjoint operator.

85

Section 3.3 Domain decomposition methods

3.3.2 Model boundary value problems Let us consider a model boundary value problem for a parabolic second-order equation. In a bounded domain , the unknown function u.x, t / satisfies the equation   m @u @u X @  k.x/ D f .x, t /, x 2 , 0 < t  T , (3.91) @t ˛D1 @x˛ @x˛ where k.x/  > 0, x 2 . The equation (3.91) is supplemented with homogeneous Dirichlet boundary conditions u.x, t / D 0,

x 2 @,

0 < t  T.

(3.92)

In addition, we specify the initial data u.x, 0/ D u0 .x/,

x 2 .

(3.93)

Thus, the time-dependent diffusion problem (3.91)–(3.93) is considered on the set of the functions u.x, t / satisfying the boundary conditions (3.92). Instead of (3.91), (3.92), we will study the operator-differential equation du C Au D f .t /, 0 < t  T . dt The Cauchy problem is investigated for the evolutionary equation (3.94): u.0/ D u0 .

(3.94)

(3.95)

For the diffusion operator, suppose

  m X @ @u k.x/ . Au D  @x˛ @x˛ ˛D1

On the set of functions (3.92), we define the Hilbert space H D L2 ./ with the scalar product and norm Z u.x/v.x/d x, kuk D .u, u/1=2 . .u, v/ D 

In H , the diffusive transport operator A is self-adjoint and positive definite: A D A  ıE,

ı D ı./ > 0,

(3.96)

where E stands for the identity operator in H . For the solution of the problem (3.94)–(3.96), we have an elementary a priori estimate, which will serve as a checkpoint in considering discrete problems. The selfadjoint and positive definite operator D is associated with the Hilbert space HD , where the scalar product and norm are defined as .u, v/D D .Du, v/,

1=2

kukD D .u, u/D ,

86

Chapter 3 Operator splitting

respectively. Multiply equation (3.94) scalarly by Au in H . In view of (3.96), we obtain the equality 1d (3.97) kuk2A C kAuk2 D .f , Au/. 2 dt Taking into account 1 .f , Au/  kAuk2 C kf k2 , 4 from (3.97), we have 1 d kuk2A  kf k2 . dt 2 In view of Gronwall’s lemma, we obtain the desired estimate Z 1 t 2 0 2 kf . /k2 d , (3.98) kukA  ku kA C 2 0 which expresses the stability of the solution of the problem (3.94)–(3.96) with respect to the initial data and the right-hand side. In addition to the parabolic equation (3.91), we consider the hyperbolic equation   m @2 u X @ @u  k.x/ D f .x, t /, x 2 , 0 < t  T (3.99) @t 2 ˛D1 @x˛ @x˛ with the boundary condition (3.92). The equation (3.99) is supplemented with two initial conditions @u (3.100) .x, 0/ D v 0 .x/, x 2 . @t The problem (3.92), (3.99), (3.100) is associated with the following Cauchy problem for the evolutionary equation of second order: u.x, 0/ D u0 .x/,

d 2u C Au D f .t /, 0 < t  T , dt 2 du u.0/ D u0 , .0/ D v 0 . dt Scalar multiplication of equation (3.101) by Adu=dt yields  2     1d   du  C kAuk2 D f , A du . 2 dt  dt A dt For the right-hand side, we use the estimate  2   1 du  du  C 1 kf k2 .   f ,A A  dt 2 dt A 2 This yields the inequality d kuk2  kuk2 C kf k2A , dt

(3.101) (3.102)

87

Section 3.3 Domain decomposition methods

where kuk2

 2  du  2  D  dt  C kAuk . A

The desired a priori estimate   Z t 2 0 2 0 2 2 ku.t /k  exp.t / kAu k C kv kA C exp. /kf . /kA d

(3.103)

0

expresses stability with respect to the initial data and the right-hand side of the Cauchy problem for the operator-differential equation (3.101).

3.3.3 Standard finite difference approximations To study approximations in space and time, we consider as an example the boundary value problems in a rectangle  D ¹ x j x D .x1 , x2 /, 0 < x˛ < l˛ , ˛ D 1, 2º. By definition, the approximate solution is given at the nodes of a uniform rectangular grid in : !N D ¹x j x D .x1 , x2 /,

x ˛ D i˛ h ˛ ,

i˛ D 0, 1, : : : , N˛ ,

N˛ h˛ D l˛ º

and let ! be the set of interior grid points (!N D ! [ @!). For the grid functions y.x/ D 0, x 2 @!, we define the Hilbert space H D L2 .!/ with the scalar product and norm X y.x/w.x/h1 h2 , kyk D .y, y/1=2 . .y, w/ D x2!

Assume that the coefficient k.x/ in the domain  is appropriately smooth, then the discrete operator of diffusion is taken in the form 1 k.x1 C 0.5h1 , x2 /.y.x1 C h1 , x2 /  y.x1 , x2 // h21 1 C 2 k.x1  0.5h1 , x2 /.y.x1 , x2 /  y.x1  h1 , x2 // h1 1  2 k.x1 , x2 C 0.5h2 /.y.x1 , x2 C h2 /  y.x1 , x2 // h2 1 C 2 k.x1 , x2  0.5h2 /.y.x1 , x2 /  y.x1 , x2  h2 //. h2

Ay D 

(3.104)

In H , the operator A is self-adjoint and positive defined [131]: A D A  .ı1 C ı2 /E,

ı˛ D

4

h˛ sin2 , h2˛ 2l˛

˛ D 1, 2.

(3.105)

88

Chapter 3 Operator splitting

The approximation of (3.91), (3.92) in space yields the semi-discrete (continuous in time but discrete in space) equation dy C Ay D f .x, t /, dt

x 2 !,

0 < t  T.

(3.106)

By (3.93), equation (3.106) is supplemented with the initial data written as y.x, 0/ D u0 .x/,

x 2 !.

(3.107)

The solution of the semi-discrete Cauchy problem (3.106), (3.107) satisfies the a priori estimate (see (3.98)) Z t 2 2  ku0 kA C kf . /k2 d . (3.108) kykA 0

Similarly, approximation in space leads us from (3.92), (3.99), (3.100) to the problem d 2y C Ay D f .x, t /, x 2 !, 0 < t  T , dt 2 dy .x, 0/ D v 0 .x/, x 2 !. y.x, 0/ D u0 .x/, dt A discrete analog of (3.103) is the estimate   Z t 2 0 2 0 2 2 exp. /kf . /kA d , ky.t /k  exp.t / kAu k C kv kA C

(3.109) (3.110)

(3.111)

0

where kyk2

 2  dy  2  D  dt  C kAyk . A

The emphasis is now on approximation in time. To construct domain decomposition methods for the problem (3.106), (3.107), we start with the usual two-level schemes. Let  be a constant time step and let y n D y.t n /, t n D n , n D 0, 1, : : : , N0 , N0  D T . The equation (3.106) is approximated by the two-level scheme with weights y nC1  y n C A.y nC1 C .1   /y n / D ' n , 

n D 0, 1, : : : , N0  1,

(3.112)

where, e.g., ' n D f . t nC1 C .1   /t n /. It is supplemented by the initial condition in the form (3.113) y 0 D u0 . The difference scheme (3.112), (3.113) has the truncation error O. 2 C .  1=2/ C h2 /, where h2 D h21 C h22 .

Section 3.3 Domain decomposition methods

89

Theorem 3.10. The scheme .3.112/, .3.113/ is unconditionally stable under the constraint   1=2, and the difference solution satisfies the estimate  2 2  ky n kD C k' n k2 , n D 0, 1, : : : , N0  1, ky nC1 kD 2

(3.114)

  1 A2 . D DAC   2

where

Proof. Let us rewrite the scheme (3.112) as   nC1   y y nC1 C y n  yn 1 A CA D ' n, EC  2  2 and multiply it scalarly by A.y nC1 C y n /. Using the fact that   1=2, and hence the operator D  A, we obtain  2 2  ky n kD C kA.y nC1 C y n /k2 D  .' n , A.y nC1 C y n //. ky nC1 kD 2 Taking into account 1 1 .' n , A.y nC1 C y n //  kA.y nC1 C y n /k2 C k' n k2 , 2 2 we get the desired estimate (3.114). The priori estimate (3.114) for the solution of the difference problem (3.112), (3.113) is a discrete analog of the a priori estimate (3.108) for the solution of the semi-discrete problem (3.106), (3.107) (D D A C O. /). To solve numerically the problem (3.109), (3.110), it is natural to use three-level schemes of the second-order approximation in time. Suppose y nC1  2y n C y n1 C A.y nC1 C .1  2 /y n C y n1 / 2 n

D' ,

(3.115)

n D 1, 2, : : : , N0  1,

where, e.g., ' n D f .t n /. In view of (3.110), for the solutions of equation (3.109), we can approximate the initial conditions as follows: y 0 D u0 ,

y1  y0  D v 0 C .' 0  Au0 /.  2

(3.116)

The truncation error of the scheme (3.115), (3.116) is O. 2 C h2 /. Theorem 3.11. The scheme .3.115/, .3.116/ is unconditionally stable under the constraint   1=4, and the difference solution satisfies the estimate S nC1  exp. /S n C

2 exp. / 2 , n D 0, 1, : : : , N0  1, k' n kA 2 exp.0.5 /  1

(3.117)

90 where

Chapter 3 Operator splitting

 n  n   n1 2  y  y n1 2   C A y C y  , Sn D       2 D   1 2 2 D DAC    A . 4

Proof. Let us introduce notation n D

y n C y n1 , 2

n D

y n  y n1 . 

In view of identities 1 1 y n D .y nC1 C 2y n C y n1 /  .y nC1  2y n C y n1 /, 4 4 y nC1 C .1  2 /y n C y n1 D y n C  .y nC1  2y n C y n1 /, we rewrite (3.115) as  nC1     1 2  nC1 C  n  n EC   A CA D ' n. 4  2

(3.118)

Multiply (3.118) scalarly in H by 2A. nC1   n / D A.nC1 C n /. In view of the notation for   1=4, we obtain S nC1  S n D  .' n , A.nC1 C n //.

(3.119)

Using the estimates for the right-hand side:   2 2 A.' n , .nC1 C n //  knC1 C n kA C "k' n kA , 2" 2 2 2 2  2.knC1 kA C kn kA /, knC1 C n kA with some " > 0, from (3.119), we derive        nC1 2  1C . (3.120) S S n C "k' n kA 1 " " 2 We choose " so that  1  D exp.0.5 /, " and therefore  1 C < exp.0.5 /. " Thus, from (3.120), we obtain the desired levelwise estimate for stability (3.117) of the difference solution. The estimate (3.117) may be treated as the direct discrete analog of the a priori estimate (3.111).

91

Section 3.3 Domain decomposition methods

Figure 3.1. Domain decomposition.

3.3.4 Domain decomposition In discussing domain decomposition methods for problems of type (3.91)–(3.93) ((3.92), (3.99), (3.100)) we usually distinguish two cases – with and without overlap. Methods with non-overlapping subdomains are associated with the explicit formulation for conditions on their common boundaries. In our case, we do not formulate an independent problem on interfaces, but we employ the corresponding exchanges of information. For methods of domain decomposition, the fundamental issue is how to organize exchanges of information between separate subdomains. In this regard, we are oriented to conventional explicit schemes. In this case, a domain decomposition may be attributed to separate subsets of grid points !˛ , ˛ D 1, 2, : : : , p: !D

p [

!˛ ,

!˛ D ¹x 2 !, x 2 ˛ º,

˛ D 1, 2, : : : , p.

˛D1

In the case of (3.104) (the 7-point stencil in space), the transition to a new time level by the explicit scheme for finding the approximate solution on the grid !˛ , ˛ D 1, 2, : : : , p is based on using the values of the solution at points that are adjacent to interfaces – we need to exchange data  @!˛ , ˛ D 1, 2, : : : , p. For the model problems considered here in a rectangle, the simplest decomposition in one spatial direction into four subdomains is shown in Figure 3.1 with overlap of subdomains. Disconnected subdomains may be treated as a single subdomain, and the

92

Chapter 3 Operator splitting

partition depicted in Figure 3.1 may be represented as the partition into two subdomains that is described by two functions ˛ D ˛ .x1 /, ˛ D 1, 2. To solve numerically the problem (3.106), (3.107), two possibilities with the minimal overlap of subdomains can be considered. The first approach employs a partition conducted through integer nodes – interface nodes belong to several subdomains (to two subdomains in our case of the partition in one variable). The second possibility uses interfaces passing through half-integer grid points of the corresponding variable. The partition through integer nodes is depicted in Figure 3.2. Assume that domain decomposition is conducted with respect to the variable x1 , i.e.,  D x1 . The partition of the domain is passed through the node  D i . Operators of decomposition are constructed, e.g., on the basis of (3.89). In view of (3.104), the decomposition operators (A1 in the subdomain 1 and A2 in the subdomain 2 , respectively) take the form A1 y D

1 k.x1  0.5h1 , x2 /.y.x1 , x2 /  y.x1  h1 , x2 // h21 1  2 k.x1 , x2 C 0.5h2 /.y.x1 , x2 C h2 /  y.x1 , x2 // 2h2 1 C 2 k.x1 , x2  0.5h2 /.y.x1 , x2 /  y.x1 , x2  h2 //, 2h2 1 k.x1 C 0.5h1 , x2 /.y.x1 C h1 , x2 /  y.x1 , x2 // h21 1  2 k.x1 , x2 C 0.5h2 /.y.x1 , x2 C h2 /  y.x1 , x2 // 2h2 1 C 2 k.x1 , x2  0.5h2 /.y.x1 , x2 /  y.x1 , x2  h2 //, 2h2

A2 y D 

x1 D  i .

Such a decomposition may be associated with Neumann boundary conditions that are used to exchange data on interfaces. The relationship between the individual subdomains is minimal, and, moreover, is restricted by exchanges of information at  D i . The values ˛ .x1 ˙ 0.5h1 , x2 /, ˛ .x1 , x2 ˙ 0.5h1 /, ˛ D 1, 2 equal 0 or 1. The second possibility deals with the partition passing through half-integer nodes, as illustrated in Figure 3.3. At the node  D i , the difference approximation has half as much flux as in the integer case. For this half-integer partition in x1 , the operators

η1 (θ)

Ω1

θi−1 θi−1/2 Figure 3.2. Partition through integer nodes.

Ω2

θi

η2 (θ)

θi+1/2 θi+1

93

Section 3.3 Domain decomposition methods

of decomposition have the form A1 y D

1 k.x1  0.5h1 , x2 /.y.x1 , x2 /  y.x1  h1 , x2 // 2h21 1  2 k.x1 , x2 C 0.5h2 /.y.x1 , x2 C h2 /  y.x1 , x2 // 4h2 1 C 2 k.x1 , x2  0.5h2 /.y.x1 , x2 /  y.x1 , x2  h2 //, 4h2 1 k.x1 C 0.5h1 , x2 /.y.x1 C h1 , x2 /  y.x1 , x2 // h21 1 C 2 k.x1  0.5h1 , x2 /.y.x1 , x2 /  y.x1  h1 , x2 // 2h1 3  2 k.x1 , x2 C 0.5h2 /.y.x1 , x2 C h2 /  y.x1 , x2 // 4h2 3 C 2 k.x1 , x2  0.5h2 /.y.x1 , x2 /  y.x1 , x2  h2 //, 4h2

A2 y D 

x1 D  i .

Calculations in the domain 1 (see Figure 3.3) involve adjacent data from the domain 2 , i.e., values at the node  D i . Thus, in such a domain decomposition, exchanges of information are minimal and coincide with the exchanges occurring in the implementation of the explicit scheme. The above variants of domain decomposition correspond to the minimal overlap of subdomains. At the discrete level, the width of overlap is determined by the mesh size (h and 2h, respectively). Similar variants can be designed for a larger overlap of subdomains (see Figure 3.4). In this case, it is evident that the costs of data exchanges increase, but a passage from one subdomain to another becomes smoother. The latter allows to expect a higher accuracy of the approximate solution. We also emphasize a special class of domain decomposition methods – the so-called substructuring methods. At the discrete level, we define inside the domain a set of subdomains and interface nodes and then we solve subproblems separately. At the continuous level, this decomposition is attributed to the subdomains with the width equal to the corresponding step of discretization in space.

η1 (θ)

Ω1

Ω2

θi−3/2 θi−1 θi−1/2 Figure 3.3. Partition through half-integer nodes.

η2 (θ)

θi

θi+1/2

94

Chapter 3 Operator splitting

η1 (θ)

Ω1

θi−1 θi−1/2

Ω2

θi

η2 (θ)

θi+1/2 θi+1

Figure 3.4. Partition through integer nodes with the overlap width 2h.

For our model discrete problem considered in a square with a square grid (h D h1 D h2 ), the computational grid ! is divided into square subdomains via a coarse grid with a step size b h. The boundaries of the subdomains (the straight lines) consist of the nodes of the computational grid. Denote this set of interior boundary nodes as b !. A fragment of the grid is shown in Figure 3.5. Such a partition of the fine computational grid corresponds to the domain decomposition presented in Figure 3.6, namely,  D 1 [ 2 , 12 D ¿. The subdomain 2 is a wireframe, and the width of the individual edges of this wireframe is equal to h. The domain 1 consists of disconnected separate subdomains. With the partition of unity (3.85), (3.86), we associate the corresponding additive representation of the identity operator E in the space H of grid functions defined on the set of internal nodes of !. Let p X

˛ D E,

˛  0,

˛ D 1, 2, : : : , p.

(3.121)

˛D1

Similarly to (3.88), the operators of decomposition can be given in the form A˛ D ˛ A,

˛ D 1, 2, : : : , p.

(3.122)

In view of (3.121), this splitting gives us the following additive representation for the problem operator: p X A˛ . (3.123) AD ˛D1

The splitting (3.123) allows us to go from equation (3.106) to the equation X dy A˛ y D f .x, t /, C dt ˛D1 p

x 2 !,

0 < t  T.

(3.124)

The direct construction of various splitting schemes for the problem (3.107), (3.124) is complicated by the fact that the individual operator terms A˛ , ˛ D 1, 2, : : : , p do not inherit the basic properties of the operator A, i.e., its self-adjointness and nonnegativity. However, using the decomposition operators (3.122), equation (3.124) can

95

Section 3.3 Domain decomposition methods

 h

h

Figure 3.5. Grid partitioning. h

Ω1

Ω2

Figure 3.6. Domain partitioning.

h

96

Chapter 3 Operator splitting

be easily transformed into the symmetric form. Multiplying equation (3.124) by the self-adjoint operator A, we obtain the equation p X dy Q C B AQ˛ y D Af .x, t /, dt ˛D1

x 2 !,

0 < t  T,

(3.125)

where the operators BQ D A,

AQ˛ D A˛ A,

˛ D 1, 2, : : : , p

are self-adjoint and non-negative. Moreover, we can introduce the new variables v D A1=2 y and then, instead of (3.125), we have the equation dv X Q C A˛ v D A1=2 f .x, t /, dt ˛D1 p

x 2 !,

0 < t  T,

(3.126)

with the self-adjoint and non-negative operators AQ˛ D A1=2 ˛ A1=2 ,

˛ D 1, 2, : : : , p.

Standard estimates for the solution of equation (3.126) in the norm of H (for kvk) correspond to using estimates in HA (for kykA ). This explains our in some sense unusual choice of the priori estimate (3.108) for the problem (3.106), (3.107) and the estimate (3.111) for the problem (3.109), (3.110). The particular specification of the decomposition operators of type (3.121), (3.122) is provided via the selection of the terms ˛ , ˛ D 1, 2, : : : , p. Some advanced features are discussed below, but we start with the simplest version. If we use the substructuring domain decomposition (see Figure 3.5), it is natural to put ² 1, x 2 b !, 1 .x/ D 1  2 .x/, x 2 !. (3.127) 2 .x/ D 0, x … b !, The operator A2 is attributed to the interface nodes b ! , whereas A1 deals with the internal nodes of the subdomains. The two-component splitting constructed above can be generalized to the case of general multicomponent splitting. The need for such an extension results from, in particular, calculations of conditions on the boundaries of subdomains, i.e., the solution of problems on graphs for two-dimensional problems. In the two-dimensional problems, considered here in a rectangle with rectangular grids, the numerical implementation of additive schemes does not face significant problems. However, for more general situations, e.g., for three-dimensional boundary value problems, the solution of these discrete problems may be too difficult. Such considerations lead to the construction of decomposition procedures for the set of boundary nodes of subdomains. A typical example is shown in Figure 3.7. The set of interface nodes is divided into ! m . Here the set of nodes on the boundary of two subdomains two parts, i.e., b !Db ! s [b

97

Section 3.3 Domain decomposition methods

Figure 3.7. Three-component partition without overlap of subdomains.

is denoted by b ! s (in Figure 3.7, it is depicted as ). The set of nodes that lie on the boundaries of a greater number of subdomains is designated as b ! m (in Figure 3.7, it is represented by ). Instead of the two-component splitting (3.121), (3.127), we now use the threecomponent splitting (3.121) with p D 3 and ² ² 1, x 2 b !s , 1, x 2 b ! m, 2 .x/ D 3 .x/ D 0, x … b !s , 0, x … b ! m, (3.128) 1 .x/ D 1  2 .x/  3 .x/, x 2 !. With such a decomposition, the calculations can be executed independent of each other in the different parts of the subdomain boundaries (on the set b ! s ) because of the known conditions at the nodes of the intersection (on the set b ! m ). The local evaluation of the solution on the boundary intersections introduces additional errors. To improve the accuracy of the approximate solution on the boundaries of subdomains, it is possible to apply algorithms with overlapping subdomains. Such a situation is shown in Figure 3.8. The set of boundary nodes b ! m , which lie near the ! m ¤ ¿, is indicated in this figure. With this in mind, boundary intersection and b !s \ b instead of (3.128), we set ² ² > 0, x 2 b !s , > 0, x 2 b ! m, 2 .x/ D 3 .x/ D 0, x…b !s , 0, x…b ! m, 2 .x/ C 3 .x/ D 1,

x2b !,

1 .x/ D 1  2 .x/  3 .x/,

x2!

and apply some variant of additive schemes of multicomponent splitting.

98

Chapter 3 Operator splitting

Figure 3.8. Three-component partition with overlapping subdomains.

3.3.5 Problems with non-self-adjoint operators We consider as a model problem the time-dependent convection-diffusion problem with constant (independent of time, but depending on the spatial coordinates) coefficients of diffusive and convective transports. The convective transport is written in the skew-symmetrical form. In a bounded domain , the unknown function u.x, t / satisfies the equation  m  @u @ 1X @u C .v˛ .x/u/ C v˛ .x/ @t 2 ˛D1 @x˛ @x˛   m X @ @u  k.x/ D f .x, t /, x 2 , @x˛ @x˛ ˛D1

(3.129) 0 < t  T,

where k.x/  > 0, x 2 . The equation (3.129) is supplemented with the homogeneous Dirichlet boundary condition (3.92) and the initial condition (3.93). In the operator-differential equation (3.94), we consider the operators of diffusive and convective transport separately: A D C C D. For the diffusion operator, assume that   m X @ @u k.x/ . Du D  @x˛ @x˛ ˛D1

(3.130)

99

Section 3.3 Domain decomposition methods

On the set of functions (3.92) in H D L2 ./, the diffusive transport operator D is self-adjoint and positive definite: D D D   ıE,

ı D ı./ > 0.

(3.131)

Let us define the convective transport operator C by the expression  m  @u @ 1X C .v˛ .x/u/ . Cu D v˛ .x/ 2 ˛D1 @x˛ @x˛ For any v˛ .x/, the operator C is skew-symmetric in H : C D C  .

(3.132)

In view of the representation (3.130), from (3.131), (3.132), it follows that A  ıE > 0 H . Domain decomposition methods will be associated with the partition of unity (3.85), (3.86) of the computational domain . In view of (3.85), (3.86), from (3.130), we obtain the representation AD

p X

A˛ ,

A˛ D C˛ C D˛ ,

˛ D 1, 2, : : : , p,

(3.133)

˛D1

where D˛ u D  C˛ u D

  m X @u @ k.x/˛ .x/ , @x˛ @x˛ ˛D1

 m  1X @u @ C .v˛ .x/˛ .x/u/ . v˛ .x/˛ .x/ 2 ˛D1 @x˛ @x˛

Similarly to (3.131), (3.132), we obtain D˛ D D˛  0,

C˛ D C˛ ,

˛ D 1, 2, : : : , p.

(3.134)

In view of (3.134), in the splitting (3.133), we have A˛  0,

˛ D 1, 2, : : : , p,

(3.135)

and, moreover, the self-adjoint part of the operator A is split into the sum of selfadjoint non-negative operators, whereas the skew-symmetric part is the sum of skewsymmetric operators. The diffusive transport operator D may be written as D D G G ,

G D k 1=2 grad ,

G  D  div k 1=2 ,

(3.136)

100

Chapter 3 Operator splitting

e and H e D .L2 .//p is the corresponding Hilbert space of vector where G : H ! H functions. Using this structure with the adjoint operators, for D˛ , ˛ D 1, 2, : : : , p, we get (3.137) D˛ D G  ˛ G , ˛ D 1, 2, : : : , p. In a similar manner, for C˛ , ˛ D 1, 2, : : : , p, we have the representation 1 (3.138) C˛ D .˛ C C C ˛ /, ˛ D 1, 2, : : : , p. 2 Using the operators of diffusive and convective transport in the forms (3.137), (3.138), we obtain directly the construction of the operators in the individual subdomains of the splitting (3.85), (3.86), and, moreover, it is easy to check that (3.134) is valid. A similar analysis is carried out below for the operators of the problem (3.106), (3.107) after approximation in space. Let us decompose the operator A into two parts, i.e., the self-adjoint part and the skew-symmetric one: 1 1 (3.139) A D C C D, C D .A  A /, D D .A C A /. 2 2 The non-negative operator D may be represented as D D G  G,

(3.140)

e . We associate the partition of unity for the domain with the where G : H ! H e in the spaces corresponding additive representation of the identity operators E and E e H and H , respectively. Suppose p X ˛ D E, ˛  0, ˛ D 1, 2, : : : , p, (3.141) ˛D1 p X

e e ˛ D E,

e ˛  0,

˛ D 1, 2, : : : , p.

(3.142)

Similarly to (3.133)–(3.135), we apply the splitting p X A˛ , A˛  0, ˛ D 1, 2, : : : , p, AD

(3.143)

˛D1

˛D1

where A˛ D C ˛ C D˛ ,

D˛ D D˛  0,

C˛ D C˛ ,

˛ D 1, 2, : : : , p.

(3.144)

In view of (3.142), for the terms of the self-adjoint part of the operator A, we can put D˛ D G e ˛ G,

˛ D 1, 2, : : : , p.

(3.145)

The decomposition of the skew-symmetric part is based on (3.141), i.e., 1 (3.146) C˛ D .˛ C C C˛ /, ˛ D 1, 2, : : : , p. 2 This additive representation is a discrete analog of (3.137), (3.138) and is interpreted as the corresponding variant of domain decomposition.

Section 3.4 Difference schemes for time-dependent vector problems

101

3.4 Difference schemes for time-dependent vector problems Here we consider the construction of additive schemes for a special class of systems of second-order parabolic and hyperbolic equations, which are typical for electrodynamic problems. Reducing the original problem to a problem for a single unknown vector function, we obtain a time-dependent problem with a nonstandard operator for the spatial variables. In general, there is no possibility to separate simple problems for the individual components of the solution at every time level. Using as an example a three-dimensional problem in a parallelepiped with the simplest homogeneous boundary conditions, we obtain an additive representation of the problem operator that allows to go via additive schemes to a sequence of two-dimensional, discrete, and elliptic boundary value problems for individual components of the approximate solution. Additive schemes are designed on the basis of the alternating triangle method.

3.4.1 Preliminary discussions So far, two- and three-level difference schemes have been constructed and investigated for basic time-dependent problems of mathematical physics. We speak of initialboundary value problems for a second-order parabolic equation (the heat equation). In this case, the elliptic operator with respect to the spatial variables may be self-adjoint or non-self-adjoint, as in the convection-diffusion equation. Similar results can also be obtained for second-order hyperbolic equations (the wave equation). In the nonself-adjoint case, unconditionally stable schemes are most often constructed under the condition that the non-self-adjoint part of the operator is subordinate to the self-adjoint part. To solve numerically initial-boundary value problems for multidimensional PDEs, special attention is paid to the construction of additive schemes, i.e., operator-splitting schemes. The passage to a chain of simpler problems allows us to construct economical difference schemes – we have splitting with respect to the spatial variables. In a number of cases, it is useful to separate subproblems of a different nature – we speak of splitting with respect to physical processes. Regionally additive schemes (domain decomposition methods) oriented to constructing numerical algorithms for parallel computers have recently been actively discussed. Additive schemes for vector problems may be treated as a new separate class. Schemes of this type can be used to develop efficient numerical algorithms for solving systems of time-dependent PDEs, i.e., vector equations. A typical situation is the case where the individual components of the unknown vector are interconnected, and it may be too difficult to derive a simple problem for evaluating vector components at a new time level. For vector problems, the alternating triangle method proposed by Samarskii [118, 131] often makes possible the construction of additive schemes. The alternating triangle method, as a rule, is treated as an iterative method for grid equations [138], but it is

102

Chapter 3 Operator splitting

primarily an additive scheme for time-dependent problems. In the alternating triangle method, the self-adjoint operator of the problem is decomposed into two operators that are the adjoints of each other. In the case of a system of ODEs, such a decomposition corresponds to the extraction of lower and upper triangular matrices. The regularization principle and the alternating triangle method were used in [89] to construct additive schemes for a dynamic elasticity problem. A similar approach was implemented for the numerical solution of problems of an incompressible fluid with variable viscosity [211]. The main peculiarity of such problems is the fact that the equations for the separate velocity components are strongly coupled (via the leading derivatives). There is no coupling in problems with a constant viscosity that permits us to solve the equations for the velocity components independently. Here we construct additive schemes for a special class of systems of second-order parabolic and hyperbolic equations, which are typical for electrodynamic problems. The system of Maxwell equations contains the divergence and rotor operators. Reducing the original problem to a problem for a single unknown vector function, we obtain a time-dependent problem with a nonstandard operator in the spatial variables. Even for problems with homogeneous media, it is impossible to extract simple problems for separate components of the solution at every time level. Naturally, the case of inhomogeneous media is more complicated. The study is performed for a model problem in a parallelepiped with the simplest homogeneous boundary conditions. Additive schemes are constructed on the basis of the Samarskii alternating triangle method. The investigation is based on the results of the work [194].

3.4.2 Statement of the problem Consider a class of vector transient problems of mathematical physics arising in continuum electrodynamics [82]. Let E and H be the intensities of the electric and magnetic fields, respectively. In the linear approximation, we denote the permittivity and permeability at the point x D .x1 , x2 , x3 / of an inhomogeneous medium by " D ".x/ and  D .x/, respectively. The governing equations, describing the dynamics of the electromagnetic field, appear as follows: rotH D  E C " rotE D 

@E , @t

@H , @t

(3.147) (3.148)

where  D  .x/ is the conductivity of a medium. In addition, we have the equations div .H / D 0,

div ."E / D 0.

(3.149)

Numerical methods for the solution of electromagnetic problems are often constructed [9, 16, 178] on the basis of the straightforward use of the system of the first-order

Section 3.4 Difference schemes for time-dependent vector problems

103

equations (3.147)–(3.149). This pertains not only to explicit and implicit schemes [5, 91, 219], but also to the issues of constructing additive schemes (see, e.g., [161]). We deal mainly with numerical methods developed for equations of the second order with respect to the spatial variables. By eliminating the magnetic field intensity from (3.147) and (3.148), we obtain the following single equation for the electric field intensity:   @E 1 @2 E C rot rotE D 0. (3.150) " 2 C @t @t  In particular cases (e.g., a quasistationary electromagnetic field in a conductor), it is convenient to consider equation (3.150) without the first term, i.e., we have an evolutionary equation of the first order in time. The second interesting example of equation (3.150) leads to a second-order evolutionary equation without the first-order derivative with respect to time ( D 0 in equation (3.150)) – this is the case of poor conductors. Note also that if equations (3.147) and (3.148) are supplemented with appropriate initial conditions, then there is no need to write equations of type (3.149) separately; they are valid automatically. For example, this is obvious for  D 0. As typical equations reflecting the main peculiarity of our problems, which is connected with space effects, we emphasize first- and second-order evolutionary equations. In the first case, the unknown vector function u.x, t / is defined in a computational domain  as the solution of the equation @u C rot .k.x/ rot u/ D 0, @t Likewise, for the second case we have

x 2 ,

0 < t  T > 0.

(3.151)

@2 u C rot .k.x/ rot u/ D 0, x 2 , 0 < t  T . (3.152) @t 2 For equations (3.151) and (3.152), we impose the corresponding initial and boundary conditions. For (3.151), suppose u.x, 0/ D u0 .x/,

x 2 ,

(3.153)

and, for equation (3.152), we specify two initial conditions @u (3.154) .x, 0/ D u1 .x/, x 2 . @t Special attention should be given to the boundary conditions for equations (3.151), (3.152). For the original electrodynamic equations (3.147), (3.148), we have, as a rule, problems with given tangential components of the electric and magnetic fields. Therefore, we consider equations (3.151), (3.152) with the boundary condition u.x, 0/ D u0 .x/,

.u n/ D 0,

x 2 @,

where n is the outward normal on the boundary.

(3.155)

104

Chapter 3 Operator splitting

To clarify nonstandard properties of the initial-boundary value problems (3.151), (3.153), (3.155) (or (3.152), (3.154), (3.155)), we restrict our considerations to the case of equation (3.151) (or (3.152)) with a constant coefficient k D const. Let the initial condition (see (3.149)) for the problem (3.151), (3.153), (3.155) be subordinated to the condition   (3.156) div u0 D 0, x 2 . It follows readily from equation (3.151) that @.div u/ D 0, @t and therefore, by (3.156), the condition div u D 0,

x2

(3.157)

is valid at all time moments. In view of (3.157), we rewrite equation (3.151) with k D const in the form @u  k u D 0, x 2 , 0 < t  T . (3.158) @t For this parabolic equation, the boundary conditions (3.155) are not sufficient for the unique evaluation of the vector function u.x, t /, x 2 , 0 < t  T . We need to use the additional equation (3.157) and then we obtain the initial-boundary value problem with the initial and boundary conditions (3.153) and (3.155)), respectively, for the overdetermined system (3.157), (3.158).

3.4.3 Estimates for the solution of differential problems Let us also present elementary a priori estimates for the solutions of the initial-boundary value problems (3.151), (3.153), (3.155) and (3.152), (3.154), (3.155). A more comprehensive study of Hilbert spaces at the continuous and discrete levels, in connection with the operators div and rot of vector analysis, can be found in the special literature. The estimates given below serve us as a guide in the construction of discrete analogs. The scalar product and the norm in the Hilbert space L2 ./ are defined in the standard way: Z u.x/v.x/d x, kuk D .u, u/1=2 . .u, v/ D 

For vector functions u D .u1 , u2 , u3 /, v D .v1 , v2 , v3 / in L2 ./ we put .u, v/ D

3 X

.u˛ , v˛ /,

kuk D .u, u/1=2 .

˛D1

On the set of functions satisfying the boundary condition (3.155), we introduce the operator Au D rot .k.x/ rot u/ . (3.159)

Section 3.4 Difference schemes for time-dependent vector problems

105

In the coordinate form, we have     @ @u1 @ @u1 .Au/1 D  k  k @x2 @x @x3 @x  2   3  @u2 @ @u3 @ k C k , C @x2 @x1 @x3 @x1     @ @u2 @ @u2 .Au/2 D  k  k @x1 @x @x3 @x  1   3  @u1 @ @u3 @ k C k , C @x1 @x2 @x3 @x2     @ @u3 @ @u3 .Au/3 D  k  k @x1 @x @x2 @x  1   2  @ @u1 @ @u2 C k C k . @x1 @x3 @x2 @x3

(3.160)

This representation of the operator A in the analysis of the problem in a parallelepiped clarifies why it is sufficient to specify the tangential component of the solution on the boundary (the boundary condition (3.155)). Taking into account that div .v u/ D u rot v  v rot u, we obtain

Z .Au, v/ D

k rot u rot v d x. 

It follows that the operator A is self-adjoint and non-negative (A D A  0) on sufficiently smooth functions satisfying (3.155) in the space L2 ./. In view of (3.159), we rewrite the initial-boundary value problem (3.151), (3.153), (3.155) in the form of the Cauchy problem for the first-order operator-differential equation: du C Au D 0, dt

0 < t  T,

u.0/ D u0 ,

(3.161) (3.162)

for u.t / 2 L2 ./. Scalar multiplication of equation (3.161) by u yields the desired a priori estimate with respect to the initial data for problem (3.161), (3.162), i.e., ku.t /k  ku0 k.

(3.163)

106

Chapter 3 Operator splitting

The problem (3.152), (3.154), (3.155) may be represented in the form of the Cauchy problem for the operator-differential equation of second order: d 2u C Au D 0, 0 < t  T , dt 2 du .0/ D u1 . u.0/ D u0 , dt

(3.164) (3.165)

 2 du  D  dt  C .Au, u/, then, in the standard way, we obtain the a priori estimate

Assume that

kuk2

ku.t /k2  ku1 k2 C .Au0 , u0 /

(3.166)

for the solution of the problem (3.164), (3.165).

3.4.4 Approximation in space Let us restrict our consideration to the case where a computational domain  is a parallelepiped:  D ¹x j x D .x1 , x2 , x3 /, 0 < x˛ < l˛ , ˛ D 1, 2, 3º. A grid used here is uniform in each direction. For grids in separate directions x˛ , ˛ D 1, 2, 3, we use notation ! ˛ D ¹x˛ j x˛ D i˛ h˛ ,

i˛ D 0, 1, : : : , N˛ ,

N˛ h˛ D l˛ º,

where !˛ D ¹x˛ j x˛ D i˛ h˛ ,

i˛ D 1, 2, : : : , N˛  1,

N˛ h˛ D l˛ º.

For the grid in the parallelepiped , we set ! D ! 1 ! 2 ! 3 D ¹x j x D .x1 , x2 , x3 /, x˛ 2 ! ˛ ,

˛ D 1, 2, 3º,

! D !1 !2 !3 . We use the standard index-free notation of the theory of difference schemes. The set of internal grid points is denoted by !, whereas @! stands for the set of boundary nodes. The approximate solution of the problem (3.151), (3.153), (3.155) is denoted by y.x, t /, x 2 !. The approximation of (3.155) leads to the discrete boundary condition: .y n/ D 0, x 2 @!, (3.167)

Section 3.4 Difference schemes for time-dependent vector problems

107

which corresponds to the conditions y˛ .x/ D 0,

xˇ D 0,

xˇ D lˇ ,

ˇ ¤ ˛,

˛, ˇ D 1, 2, 3.

On the set of grid functions satisfying the boundary condition (3.167), we introduce a discrete analog of the operator A. Just as in the approximation of a scalar elliptic problem with mixed derivatives, we set 1 A D .AC C A /, 2

(3.168)

where AC y D ..AC y/1 , .AC y/2 , .AC y/3 /, A y D ..A y/1 , .A y/2 , .A y/3 /. By the coordinate representation (3.160) for the differential operator considered here, we put .AC y/1 D .k y1x2 /x 2  .k y1x3 /x 3 C .k y2x1 /x 2 C .k y3x1 /x 3 , .A y/1 D .k y1x 2 /x2  .k y1x 3 /x3 C .k y2x 1 /x2 C .k y3x 1 /x3 .

The other components of the vectors AC y and A y may be written in a similar way. Each of the discrete operators AC and A approximates the differential operator A with the first order h D .h21 C h22 C h23 /1=2 , while Au  Au D O.h2 /. For scalar grid functions, we introduce the Hilbert space L2 .!/, where the scalar product and norm are defined as follows: X y.x/ w.x/ h1 h2 h3 , kyk D .y, y/1=2 . .y, w/ D x2!

For vector grid functions in L2 .!/, we put .y, w/ D

3 X

.y˛ , v˛ /,

kyk D .y, y/1=2 .

˛D1

By straightforward computations, we find that the discrete operator A is a self-adjoint and non-negative operator in L2 .!/, i.e., A D A  0. In particular, self-adjointness follows from the chain of equalities ..AC y/˛ , w˛ / D .y˛ , .AC w/˛ /, ..A y/˛ , w˛ / D .y˛ , .A w/˛ /, which ensure that AC D .AC / , A D .A / .

˛ D 1, 2, 3,

108

Chapter 3 Operator splitting

3.4.5 Schemes with weights To solve numerically the semi-discrete problem dy C Ay D 0, dt

x 2 !,

0 < t  T,

(3.169)

y.x, 0/ D u0 .x/,

(3.170)

we employ the usual schemes with weights. Using a constant time increment, we introduce the time grid !  D ! [ ¹T º D ¹tn D n ,

n D 0, 1, : : : , N0 ,

N0 D T º,

and set y n D y.x, t n /, t n D n . Applying two-level schemes, equation (3.169) is approximated by the equation y nC1  y n C A. y nC1 C .1   /y n / D 0, 

n D 0, 1, : : : , N0  1,

(3.171)

where  is a numerical parameter (weight), and usually 0    1. The scheme (3.171) may be written in the canonical form of two-level schemes: B

y nC1  y n C Ay n D 0, 

n D 0, 1, : : : , N0  1

(3.172)

with the operator B D E C  A.

(3.173)

A

 0; therefore, the scheme (3.172), (3.173) is unconditionally In our case, A D stable under the constraint   1=2. In this case, the solution of the difference equation (3.171) satisfies the estimate ky n k  ku0 k,

n D 0, 1, : : : , N0  1.

(3.174)

The problem (3.152), (3.154), (3.155) is treated in a similar way. The approximation in space yields the Cauchy problem for the operator-differential equation of second order: d 2y C Ay D 0, x 2 !, 0 < t  T , (3.175) dt 2 dy (3.176) y.x, 0/ D u0 .x/, .x, 0/ D u1 .x/. dt To solve numerically the problem (3.175), (3.176), it is natural to use the scheme with weights y nC1  2y n C y n1 2 C A.1 y nC1 C .1  1  2 /y n C 2 y n1 / D 0,

(3.177)

Section 3.4 Difference schemes for time-dependent vector problems

y 0 D y 0,

y 1 D w1 .

109 (3.178)

Here the grid function w1 is determined by the choice of approximations to the initial conditions (3.176). The scheme (3.177), (3.178) may be formulated in the canonical form: y nC1  y n1 C R .y nC1  2y n C y n1 / C A y n D 0 2 with the operator B

(3.179)

1 1 C 2 EC A. (3.180) 2  2 On the basis of general results on the stability of three-level operator-difference schemes, we can establish that if 1 (3.181) 1  2 , 1 C 2  , 2 then the scheme (3.179), (3.180) is stable with respect to the initial data, and the difference solution satisfies the estimate B D  .1  2 /A,

RD

kY nC1 k  kY 1 k ,

(3.182)

where 1 kY nC1 k2 D .A.y nC1 C y n /, y nC1 C y n / C ky n  y n1 k2R 1 A . 4 4 This estimate is consistent with the estimate (3.166) derived for the solution of the differential problem.

3.4.6 Alternating triangle method The schemes (3.171) and (3.177) are not convenient for numerical implementation. This is primarily due to the fact that the approximate solution y nC1 should be evaluated at a new time level from a coupled system of grid equations for separate components of the unknown vector. Let us construct additive difference schemes, where the system is decomposed and the individual components of the vector y nC1 are found from two-dimensional, grid, and elliptic problems. In accordance with the Samarskii alternating triangle method, we use an additive decomposition of the problem operator into two operators adjoint to each other. Let us apply this approach to solve numerically the vector problems (3.161), (3.162) and (3.164), (3.165). We start with an additive representation of the operator A. We treat A as the operator matrix 1 0 A11 A12 A13 (3.183) A D @ A21 A22 A23 A . A31 A32 A33

110

Chapter 3 Operator splitting

By (3.160), we have, e.g.,

    @ @u @ @u A11 u D  k  k , @x2 @x2 @x3 @x3   @ @u k , A12 u D @x2 @x1   @ @u k . A13 u D @x3 @x1

A representation similar to (3.183) is used for the discrete operator A as well. In view of(3.168), we obtain 1 0 A11 A12 A13 (3.184) A D @ A21 A22 A23 A , A31 A32 A33 where

1 C A ˛, ˇ D 1, 2, 3. A˛ˇ D .AC ˛ˇ /, 2 ˛ˇ For the individual components, we obtain, e.g., AC 11 y D .k yx2 /x 2  .k yx3 /x 3 ,

A 11 y D .k yx 2 /x2  .k yx 3 /x3 ,

AC 12 y D .k yx1 /x 2 ,

A 12 y D .k yx 1 /x2 ,

AC 13 y D .k yx1 /x 3 ,

A 13 y D .k y3x 1 /x3 .

In the alternating triangle method, the operator A is represented in the form A D A1 C A2 , Taking into account (3.184), we get 0 1 A1 D @ 0 A2 D @

2 A11 A21

A31 1 2 A11 0 0

A1 D A2 .

0 1 A 2 22 A32 A12 1 2 A22 0

1 0 0 A, 1 A 2 33 1 A13 A23 A . 1 2 A33

(3.185)

(3.186)

To solve the problem (3.169), (3.170), we use the alternating triangle method, which may be represented in the canonical form (3.172) with B D .E C  A1 / .E C  A2 / .

(3.187)

The scheme (3.172), (3.187) approximates (3.169) with the second order in time for  D 0.5 and with the first order for  ¤ 0.5. Using the relation B D B   E C  A,

Section 3.4 Difference schemes for time-dependent vector problems

111

we can establish directly the unconditional stability of this scheme with   0.5. The implementation of the alternating triangle method is based on the successive inversion of the operators E C  A1 and E C  A2 . For  D 0.5, as a rule, the implementation involves the approximate solution at a half-step: y nC1=2  y n C A1 y nC1=2 C A2 y n D 0, 0.5 y nC1  y nC1=2 C A1 y nC1=2 C A2 y nC1 D 0, 0.5

(3.188) n D 0, 1, : : : , N0  1.

(3.189)

nC1=2

To evaluate y˛ and y˛nC1 , ˛ D 1, 2, 3, we solve separate discrete elliptic probnC1=2 lems. For example, for y1 , we have 2 1 2 nC1=2 1 nC1=2 C A11 y1 D y1n  A11 y1n  A12 y2n  A13 y3n . y1  2  2 Specific features of such problems are due to the fact that, to find separate components, we need to solve two-dimensional elliptic problems with homogeneous Dirichlet boundary conditions. In this case, the third spatial variable is treated as a parameter. nC1=2 , we obtain For example, for y1  2 nC1=2 1  nC1=2 nC1=2  y1 .k y1x2 /x 2 C .k y1x 2 /x2  4  1 nC1=2 nC1=2 .k y1x3 /x 3 C .k y1x 3 /x3 D f1n , C 4 nC1=2

y1

.x/ D 0,

x˛ D 0,

x˛ D l˛ ,

˛ D 2, 3.

Discrete problems for the other components of the approximate solution are formulated in a similar way. The second-order equation may be treated in a similar way. For numerically solving the problem (3.175), (3.176), we apply the scheme y nC1  2y n C y n1 C Ay n D 0, n D 1, 2, : : : , N0  1. 2 If we use the form (3.179), then the scheme (3.190) corresponds to 1 B D 0, R D 2 D.  D

In the alternating triangle method, we can choose the operator D as    D D E C   2 A1 E C   2 A2 . By (3.185), we have

D D D   E C   2 A.

The stability condition B  0,

1 R> A 4

(3.190)

(3.191)

112

Chapter 3 Operator splitting

for the scheme (3.179) is valid for the scheme (3.190), (3.191), based on the alternating triangle method, under the constraint   0.25. The implementation of this scheme involves the inversion of the operators E C  2 A1 and E C  2 A2 via solving the twodimensional elliptic problems for individual components of the approximate solution.

3.5 Problems of hydrodynamics of an incompressible fluid To study two-dimensional unsteady problems of a viscous incompressible fluid, the governing equations are used written both in the primitive variables velocity-pressure and in the variables stream function-vorticity. As shown here, an a priori estimate for the difference solution free of any restrictions on grid parameters does exist. If we employ the variables stream function-vorticity, then we face the problem of the vorticity evaluation on rigid walls. The primitive variables yield problems in the calculation of the pressure, e.g., the so-called checker-board effects. To resolve the problems in both formulations, we consider different variants of additive schemes. This allows us to construct the iteration-free implementation of the boundary condition for the vorticity and to obtain an elliptic discrete problem for the pressure.

3.5.1 Differential problem The stream function-vorticity formulation is in common use for the numerical simulation of the dynamics of an incompressible fluid and heat and mass transfer phenomena [35, 105, 111]. At the present time, there is a variety of numerical methods for the implementation of this approach with emphasis on approximations of convective terms, boundary conditions for the vorticity and so on. Advances in the development of computational algorithms for this formulation have been achieved not only in theoretical studies, but also include a large number of numerical experiments. Let us consider an unsteady flow of an incompressible fluid in a 2D rectangular cavity under the influence of a volume distributed force. In a rectangle  D ¹x j x D .x1 , x2 /, 0 < x˛ < l˛ , ˛ D 1, 2º, the velocity vector v D .v1 , v2 / and the pressure, normalized with respect to the density and denoted by p, are governed by the momentum equation: 1 @v C V .v/v C grad p 

v D f .x, t /, @t Re x 2 , t > 0,

(3.192)

where Re is the Reynolds number (a nondimensional parameter of the problem), f .x, t / denotes the volumetric force vector, V D V .v/ stands for the convective transport operator, and is the two-dimensional Laplace operator. The equation

Section 3.5 Problems of hydrodynamics of an incompressible fluid

113

(3.192) is supplemented with the incompressibility constraint: div v D 0,

x 2 ,

t > 0.

(3.193)

The cavity boundary @ is assumed to be rigid and fixed, and so the no-slip and no-permeability boundary conditions are imposed: v.x, t / D 0,

x 2 @,

t > 0.

(3.194)

The quiescent state is considered as the initial condition: v.x, 0/ D 0,

x 2 .

(3.195)

The problem (3.192)–(3.195) provides the complete description of fluid motion in  at any time moment t > 0. To construct a numerical algorithm, we introduce transformed variables, i.e., the stream function and vorticity. The velocity components are expressed in terms of the stream function .x, t / as follows: @ @ , v2 D  , (3.196) @x2 @x1 and therefore the incompressibility constraint (3.196) holds identically. Next, for the vorticity, we have @v2 @v1  . (3.197) wD @x1 @x2 In view of (3.196) and (3.197), the momentum equation (3.192) yields the following vorticity transport equation: 1 @w C V .v/w 

w D '.x, t /, @t Re (3.198) v1 D

x 2 ,

t > 0,

where @f2 @f1  . @x1 @x2 For the stream function, we obtain Poisson’s equation 'D



D w,

x 2 ,

t > 0.

(3.199)

The system of equations is complemented (see (3.194), (3.195)) by the boundary and initial conditions: @ .x, t / D 0, x 2 @, t > 0, (3.200) .x, t / D 0, @ .x, 0/ D 0,

x 2 ,

where  stands for the outer normal vector.

(3.201)

114

Chapter 3 Operator splitting

In equation (3.198), the convective term may be written in different ways using the definition of the velocity components (3.196). For example, for the convective term in the divergent form: V .v/ D

2 X

V .˛/ .v˛ /, V .˛/ .v˛ /w D

˛D1

we get V .1/ .v1 / D

@ @x1



 @ w , @x2

@ .v˛ w/, ˛ D 1, 2, @x˛

V .2/ .v2 / D 

@ @x2



 @ w . @x1

(3.202)

(3.203)

For this definition of the convective term, we have the following fundamental property: .V .v/w, / D 0.

(3.204)

For the problem (3.198)–(3.201)), in view of (3.204), we will derive an elementary a priori estimate. Let H D L2 ./ be a Hilbert space with the norm k  k and the scalar product ., /. Scalar multiplication of equation (3.198) by yields 1 1d k grad k2  kwk2 C 2kf k k grad k. 2 dt Re Thus, the desired estimate Z t kf .x.s/ds k grad .x, t /k  k grad .x, 0/k C 2

(3.205)

0

holds. The estimate (3.205), which expresses the boundedness of the solution of the problem (3.198)–(3.201), will serve us as a guide in the construction of difference schemes.

3.5.2 Discretization in space To solve numerically the problem (3.198)–(3.201), difference methods are used here. Let us introduce in  a uniform grid !N D ! C @! with constant steps h1 and h2 . On the set of the grid functions y that are equal to zero on the boundary, we define the discrete Laplace operator: ƒy D

2 X

ƒ˛ y,

ƒ˛ y D yxN ˛ x˛ ,

˛ D 1, 2.

(3.206)

˛D1

In the finite-dimensional Hilbert space H of grid functions, we define the scalar product by the expression X y.x/z.x/h1 h2 . .y, z/ D x2!

The operator ƒ is self-adjoint and positive definite in H , i.e., ƒ D ƒ > 0.

115

Section 3.5 Problems of hydrodynamics of an incompressible fluid

Approximations of convective terms should be given special attention. If we want to construct linearized schemes, then we can employ neither the divergent form (3.202), (3.203) nor the nondivergent form for the convective terms in the vorticity transport equation due to the fact that the unconditional fulfillment of a discrete analog of the property (3.204) is hardly expected. From equations (3.196), (3.197), it follows that     @ @ @ @ w  w Vw D @x1 @x2 @x2 @x1     @ @ @w @w C D @x1 @x2 @x2 @x1       1 @w @ @ 1 @w @ @ @w @w D C C C . 2 @x2 @x1 @x1 @x2 2 @x1 @x2 @x2 @x1 Introducing the vector @w @w , q2 D  , (3.207) @x2 @x1 we can rewrite the convective transport of the vorticity as 1 Q V .v/w D V.q/   .V .q/  V  .q// . (3.208) 2 In this formulation, we treat the convective transport of the vorticity as an effective transport of the stream function. By (3.202), the adjoint operator in the expression (3.208) seems like this: q D ¹q1 , q2 º,

V  .v/ D

2 X

V˛ .v˛ /,

q1 D

V˛ .v˛ /w D v˛

˛D1

@w , @x˛

˛ D 1, 2

and corresponds to the nondivergent form of the convective terms. Now with the operator formulation (3.207), (3.208), the fundamental constraint (3.204) is valid for all vectors q. Using the formulation (3.207), (3.208), we approximate the convective terms via the central differences as follows: VQ .q/ D

2 X

VQ˛ .q˛ /,

(3.209)

˛D1

where, in view of (3.207), we obtain 1 VQ1 .q1 / D .w ı ı C .w ı / ı /, x2 x1 2 x2 x1 1 VQ2 .q2 / D  .w ı ı C .w ı / ı /. x1 x2 2 x1 x2 For this approximation of convective transport, we have .VQ .q/ , / D 0

(3.210)

(3.211)

116

Chapter 3 Operator splitting

that means that the operator VQ is skew-symmetric in H , i.e., VQ D VQ  . Moreover, both operators VQ˛ .q˛ /, ˛ D 1, 2 are also skew-symmetric. Below, We will use the same notation for the discrete solutions as for the exact ones. Let us approximate the boundary condition (3.200) using the Thom formula for the vorticity such that, e.g., on the left boundary of the domain, we get 2 w.0, x2 / D  2 .h1 , x2 /. h1 The discrete problem is written at the internal grid points, and so the discretization of equations (3.198), (3.199) in space yields the following semi-discrete (continuous in time but discrete in space) equation: d ƒ  VQ .q/ dt x 2 ,

C

1 2 ƒ Re

C

1 %.x/ Re

D '.x, t /,

t > 0.

(3.212)

The grid function %.x/ in (3.212) with Thom’s approximation of the vorticity on the boundary is defined by the expression %.x/ D

2 X

%˛ .x˛ /,

˛D1

8 h˛ < x˛ < l˛  h˛ ˆ < 0, %˛ .x˛ / D 2 ˆ : 4 , x˛ D h˛ , l˛  h˛ , h˛

˛ D 1, 2.

Rewrite equation (3.212) in the following operator form, which is more convenient for further numerical implementation: d ƒ C .A1 C A2 / dt

D ',

(3.213)

where

1 2 1 ƒ , A2 D VQ .q/ C %.x/E (3.214) Re Re and E is the identity operator. Thus, in this equation, we split the operators into two parts, where the first one is associated with diffusion, whereas the second one is nothing but convection and the boundary condition. Due to the above-mentioned properties and the peculiarities of the grid function %.x/, we have that A1 D

A1 D A1 > 0,

A2  0

for any q. Elementary difference schemes for equation (3.213), (3.214) are based on the natural linearization, where the operator VQ .q/ is evaluated using the solution at the previous time level. In view of the non-negativity of the operators A˛ , ˛ D 1, 2, we can apply various splitting schemes: factorized difference schemes, schemes of summarized

Section 3.5 Problems of hydrodynamics of an incompressible fluid

117

approximation, regularized additive schemes etc.; all of them are unconditionally stable in the case of splitting into the sum of two non-negative operators. Such difference schemes [23, 190] provide an iteration-free implementation of the boundary condition for the vorticity.

3.5.3 Peculiarities of hydrodynamic equations written in the primitive variables To solve general three-dimensional problems of hydrodynamics, computational algorithms for the Navier–Stokes equations are constructed using the standard primitive variables velocity-pressure. The main bottleneck in the numerical solving of such problems is associated with the calculation of the pressure – there is no boundary value problem for the pressure in the original problem formulation. On the basis of splitting with respect to the physical processes, where the transport due to the pressure is treated separately, it is possible to obtain efficient numerical algorithms with an elliptic boundary value problem that is formulated for the pressure at the discrete level. For the sake of simplicity, we confine ourselves to a two-dimensional unsteady problem that is discretized on a non-staggered (collocated) grid, where all unknowns are defined on a single uniform rectangular grid. Let us consider the model problem (3.192)–(3.195). The convective transport operator V may be written in the skew-symmetric form:  1 .w  grad/v C div.wv/ . V .w/v D 2 The equations (3.192), (3.193) are supplemented with the following constraint for the unique determination of the pressure: Z p.x, t /d x D 0, 0 < t  T . 

The primary bottlenecks in solving the problem (3.192)–(3.195) result from the non-self-adjointness of the problem – especially from the non-self-adjointness of the convective transport operator – and difficulties related to the calculation of the pressure [111,179]. To describe ways for how to resolve the first problem, it is convenient to consider model steady-state and unsteady problems for the convection-diffusion equation. To obtain a “good” problem for the pressure, we employ schemes of splitting with respect to the physical processes [13, 24, 52, 75, 168, 208] – analogs of classical additive difference schemes for multidimensional transient problems. In constructing discrete analogs of PDEs arising from problems of mathematical physics, the emphasis is on obtaining approximations that preserve fundamental properties of the equations under consideration at the discrete level. First of all, it is connected with monotonicity-preserving approximations of the convective-diffusive transport (see, e.g., [13,131,153]). Next, conservatism of difference schemes should be

118

Chapter 3 Operator splitting

mentioned; this means that the corresponding conservation law is satisfied at the discrete level. For the Navier–Stokes equations, the principal issue is associated with the fulfillment of the conservation law for the kinetic energy. In particular, in the momentum equation, the terms of convective transport and the pressure gradient contribute no disturbance to the kinetic energy – we say that such a scheme is energy neutral. In this case, the kinetic energy changes only due to viscous dissipation. To discretize the Navier–Stokes equations, precisely such energy neutral approximations will be applied to the convective transport and transport due to the pressure. In the numerical simulation of incompressible flows, many problems are connected with the complexity of calculating the pressure. To obtain an equation for the pressure, we construct an unconditionally stable (for linear problems) scheme via splitting with respect to the physical processes, where the transport due to the pressure is treated separately. In this case, the approximate solution satisfies an a priori estimate that is consistent with the corresponding estimate for the solution of the differential problem with any steps in time and in space. For the pressure, we obtain a completely definite discrete elliptic problem that has no problem connected with solvability and specification of boundary conditions. Considering the above-mentioned properties of numerical algorithms, we should take into account structures of computational grids. Various types of grids are in common use for the Navier–Stokes equations: non-staggered (collocated), partially staggered (ALE-type), and staggered (MAC-type) grids. To select an appropriate arrangement of unknowns, both a theoretical analysis and numerical experiments should be involved.

3.5.4 A priori estimate for the differential problem The solution of the problem considered here satisfies some a priori estimates. We will highlight the basic and elementary estimate, which from a mathematical viewpoint expresses an estimate of the solution norm, and, in the physical sense, specifies the law of change for the kinetic energy. Introduce the Hilbert space H D L2 ./ of functions with the scalar product Z .u, w/ D u.x/w.x/d x , u.x/, w.x/ 2 H 

and the corresponding norm kuk. For the vectors u, we define the Hilbert space H 2 as the direct sum H 2 D H ˚ H with the scalar product and norm .u, w/ D

2 X

.u˛ , w˛ /,

kuk D .u, u/1=2 ,

˛D1

respectively. Let H2 be a subspace of H 2 that consists of solenoidal functions, i.e., the functions satisfying the incompressibility constraint (3.193).

Section 3.5 Problems of hydrodynamics of an incompressible fluid

119

Rewrite equations (3.192), (3.193) in this subspace as a single equation in the following operator form. The operator formulation of the problem (3.192)–(3.195) is treated as the Cauchy problem for operator-differential equation: dv C V .v/v C P v C N v D f , dt

0 0.

To obtain the basic a priori estimate for the Cauchy problem (3.195), (3.215), multiply equation (3.215) by v scalarly in H2 . In view of the above-mentioned properties of the operators, we get the a priori estimate: Z t kf .x, s/kds . (3.216) kv.x, t /k  kv0 .x/k C 0

It provides the boundedness of the solution. For linear problems, such an estimate implies the solution stability with respect to the initial data and the right-hand side. In the nonlinear case, which occurs in our consideration, the estimate (3.216) ensures the stability of the trivial solution only. Therefore, this estimate is referred to as the stability estimate in the linear sense.

3.5.5 Approximation in space We will construct discrete analogs of the operators V , P , N that do have the properties mentioned above. Let us consider a uniform rectangular grid with steps h1 and h2 in a rectangle  D ¹x j x D .x1 , x2 /, 0 < x˛ < l˛ , ˛ D 1, 2º. Suppose !N D ! [ @!, where ! is the set of interior nodes and @! stands for the set of boundary nodes. Introduce the Hilbert space H of grid functions such that X y.x/z.x/h1 h2 . .y, z/ D x2!

120

Chapter 3 Operator splitting

In H 2 D H ˚ H we have .y, z/ D

2 X

.y˛ , z˛ /.

˛D1

Let ƒ be the discrete Laplace operator on the standard 5-point stencil: ƒy D 

2 X

yxN ˛ x˛ .

˛D1

The operator ƒ is self-adjoint and positive definite in H on the set of grid functions equal to zero on @!, i.e., ƒ D ƒ > 0. We define the space H2 as a subspace of H 2 that consists of solenoidal grid functions satisfying the following discrete analog of the incompressibility constraint: divh w D 0,

x 2 !,

(3.217)

N which will be specified later. where !  denotes a subset of the grid !, Consider the following semi-discrete problem: dw C V .w/w C P w C N w D f , w 2 H2 , 0 < t  T , dt w.x, 0/ D v0 .x/, x 2 !.

(3.218) (3.219)

Using the skew-symmetric form of the convective transport operator V , we apply the central differences for approximation: 2  1 X .w˛ y/ ı C w˛ y ı . V .w/y D x˛ x˛ 2 ˛D1

(3.220)

.V .w/y, y/ D 0

(3.221)

In this case, we have for any w, y 2 H 2 that equals zero on @!. Thus, the discrete operator of convective transport inherits the skew-symmetric property of the differential operator V .v/. Consider the discrete operator N such that 1 ƒ, (3.222) N D Re where ƒ is defined above. This operator inherits the properties of the operator N , i.e., N D N  > 0. It remains to construct a skew-symmetric discrete operator for P . We consider the following representation: P w D gradh p, w 2 H2

(3.223)

121

Section 3.5 Problems of hydrodynamics of an incompressible fluid

for the discrete operator P . The operator P is skew-symmetric if and only if .P w, w/ D 0, w 2 H2 .

(3.224)

To ensure the skew-symmetry of the operator P that satisfies (3.223), we must employ consistent approximations of the gradient and divergence operators. The consistency of the approximations of the operators gradh and divh is treated as the validity of the equality (3.225) .gradh p, w/ D .p, divh w/ , where

.v, y/ D

X

v.x/y.x/h1 h2 .

(3.226)

x2! 

It is clear that the equality (3.224) for the operator P , expressed as (3.223), is true if and only if the discrete operators gradh and divh are consistent in the sense of the fulflilment of (3.225), (3.226) and divh w D 0 for x 2 !  . To design consistent approximations for operators gradh and divh , we first define the operator gradh and then use the equality ((3.226) to specify the operator divh and N the set of grid pints !   !. If we select the backward difference derivative for the gradient operator: gradh p D .pxN 1 , pxN 2 /, x 2 !,

(3.227)

from (3.226), it follows that .gradh p, w/ D

2 X 

2 X    pxN ˛ , w˛ D  p, .w˛ /x˛ .

˛D1

˛D1

In this case, the discrete divergence operator obtains the following representation: divh w D .w1 /x1 C .w2 /x2 , x 2 !  .

(3.228)

It follows from (3.225) and (3.228) that the pressure p and the operator divh should be defined on the same set of nodes !  , where !  D ¹x D .x1 , x2 / j x˛ D i˛ h˛ , i˛ D 0, 1, : : : , N˛  1, i1 C i2 ¤ 0º. The approximations (3.227) and (3.228) ensure the skew-symmetry of the operator P .

3.5.6 Additive difference schemes Our time-stepping technique is based on operator-splitting schemes, which allow us to formulate the problem for the pressure in the most natural way. Rewrite (3.218) in the form dw C .A1 C A2 /w D f , 0 < t  T . dt

122

Chapter 3 Operator splitting

In the present two-component splitting, we separate the pressure operator in the following way: A1 D V .w/ C N , A2 D P ,

A1 .w/ ¤ A1  0,

A2 D A2 .

Now we can apply a scheme of splitting: ADI methods, factorized schemes, locally one-dimensional techniques etc. For example, splitting via the Douglas–Rachford method yields wnC1=2  wn (3.229) C A1 .wn /wnC1=2 C A2 wn D f n ,  wnC1  wn (3.230) C A1 .wn /wnC1=2 C A2 wnC1 D f n , x 2 !.  The scheme (3.229), (3.230) is unconditionally stable (in the linear sense):   .E C A2 /wnC1   k.E C A2 /wn k C  kf n k. The equation (3.229) corresponds to the implicit implementation of convectivediffusive transport with the explicit determination of the pressure gradient: wnC1=2  wn C .V .wn / C N /wnC1=2 C gradh p n D f n , x 2 !.  The second step (equation (3.230)) realizes the implicit transport due to the pressure gradient: wnC1  wn (3.231) C .V .wn / C N /wnC1=2 C gradh p nC1 D f n , x 2 !,  (3.232) divh wnC1 D 0, x 2 !  . Calculations in this scheme are implemented in the following way. First, solving the non-self-adjoint elliptic problem, we determine wnC1=2 . Secondly, we solve the self-adjoint elliptic problem for ıp D p nC1  p n , i.e., 1 divh gradh ıp D divh vnC1=2 , x 2 !.  And finally, this pressure correction ıp is used to obtain the solenoidal velocity field at the new time level: wnC1 D wnC1=2   gradh .p nC1  p n /,

x 2 !.

Stencils of the operator divh gradh depend on the type of the computational grid and approximations of the operators div and grad. Similarly, using schemes with splitting by physical processes, we can implement other approximations to the operators of the gradient and divergence as well as other types of computational grids. Special attention should be given to schemes based on splitting with respect to spatial variables. Some approaches to constructing locally one-dimensional schemes for problems of an incompressible fluid are discussed in the works [53, 199].

Chapter 4

Additive schemes of two-component splitting

Additive schemes, where the operator of a problem is split into the sum of two operators, are in common use for constructing numerical methods. Alternating direction implicit (ADI) methods provide classical examples of such schemes. Below, we will discuss at the operator level the issues of the construction and study of additive schemes that are similar to ADI methods. More generally, we deal with factorized additive difference schemes. We emphasize among them the class of additive alternating triangle schemes for problems with a self-adjoint operator. Possibilities of developing additive difference schemes for evolutionary equations of second order are discussed along with their generalizations to the case of multicomponent splitting schemes.

4.1 Alternating direction implicit schemes We study additive schemes of two-component splitting, which were proposed in the mid-1950s for the numerical solution of two-dimensional parabolic problems. We focus on an additive representation of a problem operator as the sum of two pairwise noncommutative operators. We establish conditions for stability and convergence of ADI schemes.

4.1.1 Problem formulation In a finite-dimensional Hilbert space H , we solve the problem du C Au D f .t /, t > 0, dt u.0/ D u0 .

(4.1) (4.2)

For simplicity, we restrict our consideration to a positive constant operator A. The schemes considered here belong to the class of additive schemes. They are characterized by the fact that the governing discrete operator (in this case, we speak of the operator A) is represented as the sum of several operators with a simpler structure: AD

p X

A˛ .

˛D1

In this case, we deal with the p-component additive splitting of the operator A.

124

Chapter 4 Additive schemes of two-component splitting

Let us consider the simplest case of two-component splitting, where A D A1 C A 2 ,

A˛  0,

˛ D 1, 2.

(4.3)

Thus, with this splitting, we assume that both operators are non-negative. We try to solve numerically the problem (4.1), (4.2) using a scheme that has the computational costs equivalent to the Cauchy problems for the equations du˛ C A˛ u˛ D f˛ .t /, ˛ D 1, 2, dt i.e., via solving some simpler (in some sense) problems.

t > 0,

4.1.2 The Peaceman–Rachford scheme To solve numerically equation (4.1), we apply to its right-hand side the following additive representation: (4.4) ' n D '1n C '2n . The classical Peaceman–Rachford scheme [104]) for the problem (4.1)–(4.4) involves two steps. First, using the known yn , we determine an auxiliary grid function, denoted by y nC1=2 , from the equation y nC1=2  y n C A1 y nC1=2 C A2 y n D 2'1n . 0.5

(4.5)

Treating y nC1=2 as the solution at the time level t D t nC1=2 , it is easy to see that (4.5) with 2'1n D ' n corresponds to the evaluation of the solution via the fully implicit scheme for the operator A1 and the explicit scheme for the operator A2 . At the second step of this ADI method, we solve the equation y nC1  y nC1=2 (4.6) C A1 y nC1=2 C A2 y nC1 D 2'2n . 0.5 Thus, the splitting with respect to spatial variables for the two-dimensional parabolic equation consists of the two steps. The first step is associated with using the fully implicit scheme for the first variable and the explicit scheme for the second one. And vice versa, the second step is based on the application of the explicit scheme for the first variable, whereas the second variable is calculated in the fully implicit way. In this interpretation, the additive scheme (4.5), (4.6) is called a scheme of ADI. The implementation of the scheme (4.5), (4.6) is based on the determination of y nC1=2 and y nC1 from the equations       E C A1 y nC1=2 D E  A2 y n C  '1n , (4.7) 2   2    E C A2 y nC1 D E  A1 y nC1=2 C  'n.2/ . (4.8) 2 2

125

Section 4.1 Alternating direction implicit schemes

The representation (4.7), (4.8) demonstrates that the splitting with respect to the spatial variables leads to the inversion of the corresponding one-dimensional discrete operator (using the Thomas algorithm); we firstly operate with one direction and then go to the transverse direction. That is why the scheme (4.5), (4.6) is sometimes referred to as the longitudinal-transverse scheme.

4.1.3 Stability of the Peaceman–Rachford scheme Here is an a priori estimate that expresses the stability of the scheme (4.5) (4.6) with respect to the right-hand side and the initial data. Theorem 4.1. Assume that operators A˛  0, ˛ D 1, 2 in the scheme .4.5/, .4.6/ are constant. Then the approximate solution satisfies the following estimate for stability with respect to the right-hand side and the initial data: n     0   nC1     X   .k'1k k C k'2k k/. (4.9)    E C A2 y  C  E C A2 y 2 2 kD0

Proof. The study is based on the inequality k.E  .1   / G/vk  k.E C   G/vk.

(4.10)

We have .E  .1   / G  /.E  .1   / G/  .E C   G  /.E C   G/ D  .G  C G/ C ..1   /2   2 / 2 G  G and therefore the inequality (4.10) holds for the operators G  0 with   0.5. From (4.7), (4.8), it follows immediately that             (4.11)  E C A1 y nC1=2    E  A2 y n  C  k'1n k, 2      2        (4.12)  E C A2 y nC1    E  A1 y nC1=2  C  k'2n k. 2 2 In view of (4.10) ( D 0.5), from (4.11), (4.12), we get              E C A2 y nC1    E C A1 y nC1=2  C  k'2n k 2 2         E  A2 y n  C  k'1n k C  k'2n k 2         E C A2 y n  C  k'1n k C  k'2n k. 2 From this inequality, we obtain the desired a priori estimate (4.9) in a usual way. We can now apply the above estimate (4.9) to the investigation of the accuracy of the Peaceman–Rachford scheme. Some issues arising in many additive schemes are concerned with the interpretation of the intermediate unknowns; in the scheme (4.5), (4.6), this is y nC1=2 .

126

Chapter 4 Additive schemes of two-component splitting

4.1.4 Accuracy of the Peaceman–Rachford scheme To investigate the accuracy of the scheme (4.5), (4.6) with '˛n D 0.5' n , ˛ D 1, 2 for the numerical solution of the problem (4.1)–(4.4), we formulate the corresponding problem for the error. Suppose, as usual, z n D y n un and let z nC1=2 D y nC1=2  uN n . The exact solution associated with y nC1=2 will be selected later. The problem for the error appears like this: z nC1=2  z n C A1 z nC1=2 C A2 z n D 0.5

n 1,

z nC1  z nC1=2 C A1 z nC1=2 C A2 z nC1 D 0.5 For the error, we have n 1

D

(4.13) n 2.

uN n  un  A1 uN n  A2 un C ' n , 0.5

unC1  uN n  A1 uN n  A2 unC1 C ' n . 0.5 Suppose, in (4.15), (4.16), that n 2

D

1  2 unC1  un uN n D .unC1 C un / C A2 . 2 4  In this case, from (4.15), (4.16), it follows that

(4.14)

(4.15) (4.16)

(4.17)

unC1  2uN n C un  A2 .unC1  un / D 0. 0.5 In addition, in view of (4.17), we get n 2

n 1

D A1



n 1

D

unC1  un  unC1  un unC1 C un  A2 u n C ' n   A2 C O. 2 /. 2  2 

Assuming ' n D f .t n C 0.5 /, for the problem solution, we get n 1

D

n 2

D O. 2 /.

(4.18)

Thus, for the above special definition of the intermediate solution (see (4.17)), the Peaceman–Rachford scheme (4.5), (4.6) with '˛n D 0.5' n , ˛ D 1, 2 has the secondorder approximation in time. To examine accuracy, we study the discrete problem (4.13), (4.14). Using the estimate (4.9) for the exact specification of the initial conditions we obtain n    nC1   X  z  .k 1k k C k 2k k/. (4.19) A  E C   2 2 kD0

Taking into account (4.18), from the estimate (4.19) it follows that the scheme converges with O. 2 / in the corresponding norm.

127

Section 4.2 Factorized schemes

4.1.5 Another ADI scheme Among other well-known ADI methods [14], in addition to (4.5), (4.6), we highlight the Douglas–Rachford scheme [28]. It may be written as y nC1=2  y n C A1 y nC1=2 C A2 y n D ' n , 

(4.20)

y nC1  y nC1=2 C A2 .y nC1  y n / D 0. 

(4.21)

In the first step (4.20), we approximate the original equation over the whole time interval; the second step (4.21) is performed to ensure stability. That is why the schemes of type (4.20), (4.21) are often called schemes of stabilizing correction. It is easy to verify that the scheme (4.20), (4.21) has the truncation error O. / and it is absolutely stable. The appropriate a priori estimate for the additive scheme (4.20), (4.21) has the form n X nC1 0 k  k.E C A2 /y k C  k' k k. k.E C A2 /y kD0

The ADI schemes considered here may be treated as a special variant of factorized schemes, which are studied below in detail. Generalizations of ADI methods to the case of multicomponent splitting (p > 2) for general pairwise noncommutative operators A˛ , ˛ D 1, 2, : : : , p face significant difficulties, despite a large number of works in this direction. Among ADI methods, we emphasize the case of the self-adjoint operator A, where, in (4.3), we have A1 D A2 . To solve numerically the Cauchy problem for systems of linear differential equations, the operators A˛ , ˛ D 1, 2 are associated with lower and upper triangular matrices. In this case, an additive scheme is referred to as an alternating triangle method. Additive schemes of this type will be studied later.

4.2 Factorized schemes We now consider possibilities of constructing a class of additive difference schemes that are based on the representation of a problem operator at the upper time level as the product of simpler operators. Two-component variants of factorized schemes are closely related to ADI methods. In designing factorized additive schemes, we apply a common approach for deriving difference schemes with a specified quality – the regularization principle for difference schemes.

128

Chapter 4 Additive schemes of two-component splitting

4.2.1 General considerations We solve the Cauchy problem for a model evolutionary equation of first order: du C Au D f .t /, dt

t > 0,

u.0/ D u0

(4.22) (4.23)

with the two-component splitting into two constant non-negative operators: A D A1 C A2 ,

A˛  0,

˛ D 1, 2.

(4.24)

For numerically solving the problem (4.22)–(4.24), we construct a two-level difference scheme, which has the canonical form: y nC1  y n (4.25) C Ay n D ' n , t n 2 ! .  A factorized scheme corresponds to the selection of the operator B in the form B

B D B1 B2 ,

(4.26)

where B˛ D E C  A˛ ,

˛ D 1, 2.

(4.27)

In such a representation of B, each of the operators B˛ , ˛ D 1, 2 corresponds to using the standard scheme with weights for an individual operator term. The computational implementation of the factorized scheme (4.24)–(4.27) involves the solution of two problems: B1 v n C Ay n D ' n , y nC1  y n D v n , t n 2 ! .  Splitting with respect to spatial variables for the two-dimensional parabolic problem results in the sequential solution of one-dimensional problems associated with the differential operator for the corresponding direction. B2

4.2.2 ADI methods as factorized schemes In some cases, ADI methods may be written as the factorized scheme (4.24)–(4.27). Consider, e.g., the scheme y nC1=2  y n C A1 y nC1=2 C A2 y n D 2'1n , 0.5

(4.28)

y nC1  y nC1=2 C A1 y nC1=2 C A2 y nC1 D 2'2n . 0.5

(4.29)

129

Section 4.2 Factorized schemes

Combining equations (4.28) and (4.29), we obtain y nC1  y n y nC1 C y n C A1 y nC1=2 C A2 D '1n C '2n .  2 Multiplying this equation from the left by the operator B1 , defined according to (4.27) with  D 0.5, we get B1 B2

y nC1  y n C A1 B1 y nC1=2 C B1 A2 y n D B1 .'1n C '2n /. 

(4.30)

For the intermediate grid functions y nC1=2 in equation (4.28), we have    B1 y nC1=2 D E  A2 y n C  '1n . 2 The substitution into (4.30) yields the factorized scheme (4.24)–(4.27) with  D 0.5, where       ' n D E  A1 '1n C E C A1 '2n . 2 2 It is easy to see that the Douglas–Rachford scheme (4.41), (4.42) may be written as (4.24)–(4.27) with  D 1.

4.2.3 Stability and accuracy of factorized schemes The factorized scheme has the canonical form of the two-level scheme (4.25) with the operators A and B determined via (4.24) and (4.26), (4.27). A direct study on stability through checking the necessary and sufficient condition is difficult because of nonself-adjointness of the operators and the nonpositivity of the operator B. Therefore, we can try to prove the stability of the scheme (4.24)–(4.27)) in more complicated norms. The proof of the stability of the ADI methods presented above serves us as an example. Theorem 4.2. Assume that in the factorized scheme .4.24/–.4.27/ the operators A˛  0, ˛ D 1, 2 are constant. Then under the constraint   0.5 this scheme is unconditionally stable and its solution satisfies the a priori estimate kB2 y nC1 k  kB2 y 0 k C

n X

 k' k k.

(4.31)

kD0

Proof. We write (4.24), (4.25) in the form B2 y nC1 D B2 y n  B11 Ay n C B11 ' n .

(4.32)

Adding to and subtracting from the right side of (4.32) the term .2 /1 B2 y n , we get B2 y nC1 D

2  1 1 B2 y n C .B2  2 B11 A/y n C B11 ' n . 2 2

(4.33)

130

Chapter 4 Additive schemes of two-component splitting

By (4.27), we obtain B2  2 B11 A D B11 .B1 B2  2 A/

D B11 ..E C  A1 /.E C  A2 /  2 A/

D

B11 .E

(4.34)

  A1 /.E   A2 /.

By definition, we put w n D B2 y n and rewrite (4.33) as w nC1 D

2  1 n 1 w C Qw n C B11 ' n . 2 2

(4.35)

In view of (4.34), the operator Q has the representation Q D .E C  A1 /1 .E C  A1 /.E   A2 /.E C  A2 /1 . Rewrite Q in the form Q D S1 S2 , where

S˛ D .E C  A˛ /1 .E   A˛ /,

(4.36) ˛ D 1, 2.

(4.37)

Let us formulate the following statement, which is known as Kellogg’s lemma. Lemma 4.1. Assume that the operator G  0 in H , then the following estimate is satisfied: (4.38) k.E  G/.E C G/1 k  1. Proof. The inequality (4.38) is equivalent to the operator inequality J D E  S  S  0, where S D .E  G/.E C G/1 . Thus, we have .E C G  /J.E C G/ D .E C G  /.E C G/  .E  G  /.E  G/ D 2.G  C G/ and therefore the inequality (4.38) holds for any non-negative operator G. By virtue of Lemma 4.1, we get kS˛ k  1, ˛ D 1, 2. In view of (4.36), the norm of the operator Q does not exceed unit, too. Thus, (4.35) implies that kw nC1 k  kw n k C  k' n k, for all   0.5. Therefore, we obtain the desired estimate (4.31) for the difference solution. On the basis of the stability estimate (4.31) we can investigate  y nC1  y n   1 y nC1 C y n EC   AC 2  2 A1 A2 CA D ' n, 2  2

t n 2 ! . (4.39)

131

Section 4.2 Factorized schemes

For the error z n D y n  un , from (4.31), the next estimate follows: kB2 z nC1 k 

n X

k

k

k,

kD0

where, in view of (4.39), the truncation error of the above scheme satisfies the representation   1 du n D  A .t n / C O. 2 /. 2 dt Thus, the factorized scheme (4.24)–(4.27) converges with the second order in  for  D 0.5, and with the first order for  ¤ 0.5.

4.2.4 Regularization principle for constructing factorized schemes To construct unconditionally stable factorized schemes, we use a common approach to improve quality of difference schemes – the regularization principle for difference schemes. Nowadays it is considered the primary methodological principle for improving difference schemes. It was proposed and implemented in a large number of examples in the work [126]. For general two- and three-level schemes, we formulate recipes for how to improve quality (stability, accuracy, efficiency) of difference schemes. Applying this principle, a wide class of difference schemes for boundary value problems of mathematical physics is studied to verify their stability and convergence [131,134]. The regularization principle for difference schemes is based on using the results that are already known for stability conditions. From this point of view, we can treat it as an element of the constructive application of general results of the stability theory. This is achieved by rewriting a scheme in a fairly general canonical form and by the formulation of stability criteria in a form that is convenient for checking. Unconditionally stable schemes are derived on the basis of the regularization principle as follows: 

a simple scheme, which is called a generating scheme, is constructed for the original problem; this scheme does not have the desired properties, i.e., it may be conditionally stable or even absolutely unstable;



this scheme is written in a unified (canonical) form, where stability conditions are known;



the quality of the scheme (its stability) is improved by perturbing its operators.

In accordance with the regularization principle, we begin with a difference scheme that will be used as the staring point. As a generating scheme for the problem (4.22), (4.23)) with A > 0, it is natural to consider the simplest explicit scheme: y nC1  y n C Ay n D ' n , 

t n 2 ! .

(4.40)

132

Chapter 4 Additive schemes of two-component splitting

Let us rewrite the scheme (4.40) in the canonical form for two-level operator-difference schemes (4.25)) with the operator B D E, where E is the identity operator. In view of the necessary and sufficient condition for stability (Theorem 2.2):  (4.41) B  A, 2 from A  kAkE we obtain the following restriction on a time step for the explicit scheme: 2 .  kAk According to (4.41), stability can be improved in two ways. In the first case, this is conducted by increasing the energy .By, y/ of the operator B (the left-hand side of (4.41)); the second way is to decrease the energy of the operator A (the right-hand side of (4.41)). Let us study the possibilities associated with the addition of operator terms to the operators B and A in order to construct unconditionally stable factorized schemes. In this case, we deal with additive regularization. The regularization is performed via the transition from the identity operator B in the scheme (4.40) to the factorized scheme (4.25), (4.26), where (4.42) B˛ D E C R˛ , ˛ D 1, 2, and R˛ , ˛ D 1, 2 stand for some operators of regularization. At the chosen regularization, we want to remain in the class of self-adjoint positive operators B. The splitting (4.24) into the self-adjoint non-negative operators A˛ , ˛ D 1, 2 yields the factorized operator B, defined by (4.26), (4.27), that is self-adjoint and positive for the commutative operators A˛ , ˛ D 1, 2. In general, the splitting (4.24) does not guarantee the commutativity of the operators. To regularize (4.42), we use the operators R˛ , ˛ D 1, 2 that satisfy the property of commutativity. Suppose R D R1 C R2 , R˛ D R˛  0, ˛ D 1, 2, (4.43) R1 R2 D R2 R1 , then

B D B  D E C R C 2 R1 R2  E C R.

Consider the regularization operator R such that R  A,

 D const > 0,

(4.44)

then the factorized scheme (4.25), (4.42), (4.43) is stable under the restriction  .  2 This follows from the basic inequality (4.41) due to the specified conditions (4.44) for the regularizer.

133

Section 4.2 Factorized schemes

In constructing factorized schemes, the most important issue is to select an appropriate regularization operator R and its decomposition (4.43). Considering difference schemes for problems of mathematical physics with variable coefficients, the standard regularizers deal with the difference operator, which corresponds to a problem with constant coefficients, and the operator terms R˛ , ˛ D 1, 2 associated with difference operators for individual variables. For problems with highly variable coefficients, the condition (4.44) may be very strong. In this way, we can design regularized factorized schemes with multicomponent splitting of a problem operator.

4.2.5 Factorized schemes of multicomponent splitting To solve three-dimensional problems or to construct general domain decomposition methods, we need to apply additive schemes of multicomponent splitting, where AD

p X

A˛ , A˛ D A˛  0, ˛ D 1, 2, : : : , p, p > 2.

(4.45)

˛D1

In constructing the factorized operator by the formula BD

p Y

.E C  A˛ /,

(4.46)

˛D1

only schemes with the pairwise commutative operators A˛ , ˛ D 1, 2, ..., p will be stable. Unfortunately, in general, we have pairwise noncommutative operators B ¤ B  even for two-component splitting. A factorized difference scheme with multicomponent splitting was proposed in the work [149]. This scheme is written in the canonical form (4.25) with the operator BD

p Y ˛D1

.E C  ApC1˛ /

p Y

.E C  A˛ /.

(4.47)

˛D1

In this case B D B  > 0, and so results of the general stability theory for operatordifference schemes can be applied. It should be noted that the representation of the operator B in the factorized form (4.46) with  D 1 makes it possible to construct unconditionally stable schemes of multicomponent splitting [93, 218]; these will be discussed below in detail. Now we present their interpretation in terms of the regularization principle for difference schemes. In numerically solving the Cauchy problem (4.22), (4.23), we consider as a producing scheme the following fully implicit scheme: y nC1  y n C Ay nC1 D ' n , 

t n 2 ! ,

134

Chapter 4 Additive schemes of two-component splitting

which is unconditionally stable. Rewrite it as .E C A/y nC1 D y n C ' n ,

t n 2 ! .

(4.48)

To obtain a more convenient implementation of the factorized scheme, in accordance with the regularization principle, we introduce small perturbations in the operator coefficients. Instead of (4.48) in the splitting (4.45), it is natural to consider the scheme p Y .E C A˛ /y nC1 D y n C ' n , t n 2 ! . (4.49) ˛D1

In fact, this corresponds to a perturbation of the operator A by a term of order O. /. The regularized scheme (4.49) approximates the original problem with the first order with respect to  . To check the stability of the factorized scheme with multicomponent splitting (4.49), we write it in the form .E C A1 /y nC1=p D y n C  ' n , .E C A˛ /y nC˛=p D y nC.˛1/=p ,

˛ D 2, 3, : : : , p,

t n 2 ! .

Hence we immediately obtain that ky nC1=p k  ky n k C  k' n k, ky nC˛=p k  ky nC.˛1/=p k,

˛ D 2, 3, : : : , p,

t n 2 ! .

Thus, we have the estimate ky nC1 k  ky n k C  k' n k,

t n 2 ! ,

ensuring the stability of the scheme (4.49) in H . Some shortcomings of the factorized scheme (4.49) are largely compensated by simplicity of its design. Among the major advantages, we emphasize the possibility of splitting into the sum of any number of noncommutative nonnegative operators A˛ , ˛ D 1, 2, : : : , p. In addition, it is possible to study such factorized schemes in Banach spaces of grid functions.

4.3 Alternating triangle method The alternating triangle method has been developed in [118] for solving the Cauchy problem for systems of linear ODEs with a symmetric matrix. It is based on splitting of the equation matrix into two triangular matrices. A general description of the alternating triangle method is presented below. The convergence of the second order with respect to time is shown for this additive scheme. Possibilities for constructing new classes of additive alternating triangle schemes are discussed with emphasis on problems with non-self-adjoint operators.

135

Section 4.3 Alternating triangle method

4.3.1 General description of the alternating triangle method Let us consider the Cauchy problem du C Au D f .t /, dt

t > 0,

(4.50)

u.0/ D u0

(4.51)

with a constant operator A D A  0. The alternating triangle method is defined by the two-component additive splitting: A D A1 C A2 > 0,

A1 D A2 .

(4.52)

Let (4.50), (4.51) be the operator formulation of the Cauchy problem for the system of linear ODEs of first order: dui .t / X aij uj .t / D fi .t /, C dt m

t > 0,

(4.53)

j D1

ui .0/ D u0i ,

i D 1, 2, : : : , m.

(4.54)

Here u D ¹u1 , u2 , : : : , um º stands for the vector of unknowns, f D ¹f1 ,f2 , : : : , fm º is a specified vector of the right-hand sides, and A D ¹aij º presents a symmetric real matrix with elements aij D aj i , i , j D 1, 2, : : : , m. Under the above conditions, the matrix A in the problem (4.53), (4.54) is treated as a self-adjoint linear operator in the finite-dimensional Hilbert (Euclidean) space H D l2 equipped with the scalar product and norm: .y, v/ D

m X

yi vi ,

kyk D .y, y/1=2 .

iD1

For the elements of the matrices ˛ º, A˛ D ¹aij

˛ D 1, 2

in correspondence with the decomposition (4.52), we have 8 8 i < j, aij , 0, i < j, ˆ ˆ ˆ ˆ ˆ ˆ < < 1 1 1 2 aij D D aij ai i , i D j , ai i , i D j , ˆ ˆ 2 2 ˆ ˆ ˆ ˆ : : i > j. 0, i > j, aij , Thus, the matrix A is split into two triangular matrices. The standard variant of the alternating triangle method employs the Peaceman– Rachford ADI technique for solving the problem (4.50)–(4.52): y nC1=2  y n C A1 y nC1=2 C A2 y n D ' n , 0.5

(4.55)

136

Chapter 4 Additive schemes of two-component splitting

y nC1  y nC1=2 (4.56) C A1 y nC1=2 C A2 y nC1 D ' n . 0.5 The implementation of the above additive scheme is connected to the consecutive inversion of the upper and the lower triangular matrices, which explains the name alternating triangle method. The ADI scheme (4.55), (4.56) satisfies (see Theorem 4.1) the a priori estimate n     0   nC1     X  y y  k' k k. A A  E C C E C     2 2 2 2 kD0

This scheme converges with the second order in  .

4.3.2 Investigation of stability and convergence Additive schemes of ADI type can be investigated in the most complete way using results of the general theory of stability for operator-difference schemes. Let us rewrite the two-level factorized scheme of the alternating triangle method in the canonical form y nC1  y n (4.57) C Ay n D ' n , t n 2 ! B  with the operator B D .E C  A1 /.E C  A2 /. (4.58) The scheme (4.57), (4.58) is equivalent to the scheme (4.55), (4.56)) if the weight parameter equals one half, i.e,  D 0.5. Theorem 4.3. The factorized scheme of the alternating triangle method .4.52/, .4.57/, .4.58/ is unconditionally stable in HA under the restriction   0.5. The following a priori estimate holds: 2 2  ky0 kA C ky nC1 kA

n 1X  k' k k2 . 2

(4.59)

kD0

Proof. We use Theorem 2.8 for (4.58) and have B D E C  A C  2  2 A1 A2 . In view of (4.52), we get B D B   E C  A. Therefore, the inequality (2.49) is satisfied for   0.5 if we choose G D E, and then the a priori estimate (2.50) takes the desired form (4.59).

Section 4.3 Alternating triangle method

137

The convergence of the factorized scheme of the alternating triangle method (4.52), (4.57), (4.58) is studied in the standard way. The equation for the error z n D y n  un has the form z nC1  z n C Az n D n , t n 2 ! , B  with the truncation error  1  du n n D  A .t / C O. 2 /. 2 dt Therefore, in view of (4.59), the scheme (4.52), (4.57), (4.58) converges in HA with the second order with respect to  for  D 0.5, and only with the first order if  ¤ 0.5.

4.3.3 Three-level additive schemes To improve the integration accuracy for systems of ODEs, it seems natural to apply multilevel difference schemes (multistep methods). Here we highlight some possibilities of constructing three-level schemes for the alternating triangle method. The process of constructing factorized schemes that are based on the alternating triangle method can be interpreted in terms of the regularization principle for difference schemes. Starting with a generating scheme of second-order accuracy, which is modified by means of small perturbations of operator coefficients, we obtain the scheme that is implemented through the inversion of discrete operators for individual operator terms in the additive representation (4.52). Let us begin with the above scheme (4.52), (4.55), (4.56). To introduce a generating scheme, we apply the Crank–Nicolson scheme y nC1 C y n y nC1  y n (4.60) CA D ' n , t n 2 ! ,  2 which is absolutely stable with second-order accuracy. It may be written in the canonical form (4.57) with  B D E C A. 2 To construct a scheme with a factorized operator, which is more convenient for numerical implementation, we perturb the operator B. Remaining in the class of unconditionally stable schemes with second-order accuracy, we put  (4.61) B D E C A C  2 A1 A2 . 2 For   0, the splitting (4.52) has  B D B   E C A. 2 The operator (4.61) may be represented in the factorized form:      B D E C A1 E C A2 , 2 2 if we select  D 0.25.

138

Chapter 4 Additive schemes of two-component splitting

Quite similarly, we can construct three-level additive schemes for the alternating triangle method. Let us consider as a producing scheme, e.g., the implicit scheme with second-order accuracy: 3y nC1  4y n C y n1 (4.62) C Ay nC1 D ' n , t n 2 ! . 2 This scheme is rewritten in the canonical form of three-level operator-difference schemes as y nC1  y n1 C R.y nC1  2y n C y n1 / C Ay n D ' n , n D 1, 2, : : : 2 with the operators 1 1 B D E C A, R D E C A.  2 The stability condition 1 R> A 4 for the scheme (4.62) is valid for all  > 0. The regularization of the scheme (4.62) is performed as follows: B

D where

3y nC1  4y n C y n1 C Ay nC1 D ' n , 2

t n 2 ! ,

D D D  D E C  2 A1 A2  E.

(4.63) (4.64)

The scheme (4.63), (4.64) with  > 0 remains absolutely stable with second-order accuracy. To prove this fact, multiply equation (4.63) from the left by D 1=2 and denote D 1=2 y n D v n . Then equation (4.63) takes the form that is quite similar to (4.62), i.e., 3v nC1  4v n C v n1 Q nC1 D 'Q n , t n 2 ! , C Av 2 where AQ D AQ D D 1=2 AD 1=2 > 0, 'Q n D D 1=2 ' n . It remains to specify the regularization parameter . To evaluate the solution at a new time level, we inverts the operator      2 2 2 4 2 D C A D E C A1 E C A2 C    A1 A2 . 3 3 3 9 If we select  D 4=9, then the regularized scheme (4.63), (4.64) becomes a factorized additive scheme. Similarly, on the basis of the regularization principle, we can construct three-level factorized schemes of the alternating triangle method, where a generating scheme is taken as follows: y nC1  y n1 C A.y nC1 C .1  2 /y n C y n1 / D ' n , t n 2 ! . 2 This scheme is stable for  > 0.25, and it converges with the second order.

139

Section 4.3 Alternating triangle method

4.3.4 Problems with non-self-adjoint operators The alternating triangle method is most suitable for constructing numerical methods to solve problems with a self-adjoint operator A. Here we discuss some possibilities to apply this approach to solving problems with non-self-adjoint operators with some restrictions on their skew-symmetric part. Suppose that, in the problem (4.50), (4.51), we have A D A0 C A1 > 0,

(4.65)

where the self-adjoint and skew-symmetric parts of A are separated: 1 1 A0 D .A C A /, A1 D .A  A /. 2 2 Assume that the skew-symmetric part is subordinated to the self-adjoint part in sense of the fulfillment of the estimation kA1 yk2  M.A0 y, y/,

M D const > 0.

(4.66)

Restrictions of this type are typical for parabolic problems with an elliptic operator of second order, where terms with the first derivatives are subordinated to terms with higher derivatives. Let us construct factorized schemes of the alternating triangle method using the splitting of the self-adjoint part of the operator A, i.e., A0 D A01 C A02 > 0,

.A01 / D A02 .

(4.67)

Remaining in the class of schemes with the second-order approximation, we focus on employing three-level factorized schemes. A reference point is the scheme y nC1  y n1 C A0 .y nC1 C .1  2 /y n C y n1 / C A1 y n D ' n , 2

t n 2 ! . (4.68)

The scheme (4.68) may be written as B

y nC1  y n1 C R.y nC1  2y n C y n1 / C A0 y n D 'Qn , 2

where B D E,

R D A0 ,

(4.69)

'Qn D ' n C A1 y n .

The scheme (4.69) with  > 0.25 satisfies (see Theorem 2.14) the a priori estimate 1 E nC1  E n C  k'Q n k2 , (4.70) 2 where 1 E nC1 D .A0 .y nC1 C y n /, y nC1 C y n / 4  1  0 nC1  y n /, y nC1  y n /. .A .y C  4

140

Chapter 4 Additive schemes of two-component splitting

In view of the subordination inequality (4.66), we have 1 n 2 k'Q k  k' n k2 C M.A0 y n , y n /. 2 It remains to estimate the last term in this inequality with respect to E n . This can be done by applying stronger restrictions to the weight: 1C" , (4.71) > 4 where " > 0. In this case (see (2.99) in the proof of Theorem 2.13), the estimate 1C" n E .A0 y n , y n /  " holds. The substitution into (4.70) yields the desired levelwise estimate for -stability: E nC1  E n C  k' n k2 ,

(4.72)

where

1C" M. " To design a factorized scheme of the alternating triangle method, we carry out the regularization of the generating scheme (4.69) via perturbations of the operator R. In view of (4.67), we set  D1C

B D E,

R D A0 C A01 A02 ,

'Qn D ' n C A1 y n .

For determining the solution at the new time level, we invert the operator 1 1 .B C 2R/ D .E C 2 A01 /.E C 2 A02 / C .  2 2 /A01 A02 . 2 2 Thus, for deriving a factorized scheme, it is sufficient to put  D 2 2 . Further arguments completely repeat the case (4.69). Thus, we get the factorized scheme y nC1  y n1 C .A0 C 2 2 A01 A02 /.y nC1  2y n C y n1 / C Ay n D ' n , 2 t n 2 ! with second-order accuracy, which is stable under the condition (4.71), and the difference solution satisfies the estimate (4.72).

4.4 Equations of second order Now we consider possibilities of constructing additive schemes to solve the Cauchy problem for an evolutionary equation of second order. Much attention is given to studying factorized difference schemes with emphasis on the alternating triangle method.

141

Section 4.4 Equations of second order

4.4.1 Model problem In a finite-dimensional Hilbert space H , we consider the Cauchy problem for the second-order equation d 2u C Au D f .t /, t > 0, dt 2 du .0/ D v 0 . u.0/ D u0 , dt

(4.73) (4.74)

Assume that the operator A is constant (independent of t ), self-adjoint, and positive (A D A > 0). In constructing difference schemes for the problem (4.73), (4.74), we are oriented to the stability estimate with respect to the initial data and the right-hand side, which is valid in the continuous case: Zt ku.t /k2



2 exp.t /.ku0 kA

C kv k C 0 2

exp. /kf . /k2 d /, 0

 2  du  2  D  dt  C kukA . For (4.73), (4.74), it is natural to use the scheme with the second-order approximation: where

kuk2

y nC1  2y n C y n1 C A.y nC1 C .1  2 /y n C y n1 / 2 n

D' , y 0 D u0 ,

(4.75)

n D 1, 2, : : : ,

y 1 D u1 .

(4.76)

This scheme may be written in the canonical form for three-level operator-difference schemes: B

y nC1  y n1 C R.y nC1  2y n C y n1 / C Ay n D ' n , 2

n D 1, 2, : : : ,

where

1 E C A. 2 Under the constraint   0.25 (see Theorem 2.16), the scheme with weights is stable. The appropriate estimate for stability with respect to the initial data (' D 0) has the form kY nC1 k  kY 1 k , B D 0,

where

RD

1 2 C ky n  y n1 k2R 1 A . kY nC1 k2 D ky nC1 C y n kA 4 4

142

Chapter 4 Additive schemes of two-component splitting

The numerical implementation of the scheme (4.75), (4.76) involves the inversion of the operator R D  2 E C A. In developing additive schemes, we separate operators of simpler structure A˛ , ˛ D 1, 2, : : : , p such that AD

p X

A˛ ,

A˛ D A˛  0,

˛ D 1, 2, : : : , p.

(4.77)

˛D1

The transition to the new time level is performed via solving p problems with the operators A˛ , ˛ D 1, 2, : : : , p.

4.4.2 Factorized schemes In designing additive schemes, we start with the simplest case of splitting into operators commutative with each other, i.e., A˛ Aˇ D Aˇ A˛ ,

˛, ˇ D 1, 2, : : : , p.

(4.78)

In this case, in the scheme D

y nC1  2y n C y n1 C Ay n D ' n , 2

n D 1, 2, : : : ,

(4.79)

we specify the operator D in the factorized form DD

p Y

.E C   2 A˛ /.

˛D1

By (4.77), (4.78), we obtain D D D   E C   2 A, and therefore the scheme (4.79) is stable for   0.25, and it converges with the second order. The requirement that the operators A˛ , ˛ D 1, 2, : : : , p are commutative is very strong; it is rare in occurrence. That is why we need to modify the factorized scheme under consideration. We discuss some basic approaches to constructing unconditionally stable factorized schemes. Assume that there exists a constant operator G such that G  A,

 D const > 0,

and it has an additive representation with pairwise commutative operators of simpler structure: p X G˛ , G˛ D G˛  0, ˛ D 1, 2, : : : , p, GD ˛D1

143

Section 4.4 Equations of second order

G˛ Gˇ D Gˇ G˛ , If we put DD

p Y

˛, ˇ D 1, 2, : : : , p.

.E C  2 G˛ /,

˛D1

then the factorized scheme (4.79) is stable under the condition 1  . 4 To numerically solve boundary value problems for parabolic equations, we select a discrete elliptic operator with constant coefficients as G. In doing so, the operators G˛ , ˛ D 1, 2, : : : , p are the difference operators with respect to various directions. More generally, we can focus on using the symmetrized factorized scheme for multicomponent splitting. Similarly to (3.78), in the scheme (4.79), we put DD

p Y

.E C   2 A.pC1˛/ /

˛D1

p Y

.E C   2 A˛ /.

(4.80)

˛D1

In this case, we have D D D  , and so the scheme (4.79), (4.80) can be investigated applying the standard criteria for stability.

4.4.3 Schemes of the alternating triangle method It is possible to construct additive schemes for solving the problem (4.73), (4.74) on the basis of the splitting A D A1 C A2 > 0,

A1 D A2 ,

(4.81)

which defines the alternating triangle method. In the scheme (4.79), we specify the factorized operator as D D .E C   2 A1 /.E C   2 A2 /.

(4.82)

In view of (4.81), we have D D D  E C   2A and therefore the factorized scheme of the alternating triangle method (4.79), (4.81), (4.82) is stable for   0.25. The investigation of the stability and convergence of the factorized scheme (4.79), (4.81), (4.82) is conducted in the standard way using common results of stability for three-level operator-difference schemes. The scheme is stable for   0.25, and it converges with the second order.

Chapter 5

Schemes of summarized approximation

Unconditionally stable additive schemes of multicomponent splitting with arbitrary noncommutative operators are based on the full approximation concept. Now we discuss approaches to the solution of intermediate problems for individual operator terms. We study schemes of componentwise splitting in the sequential and parallel versions (additively averaged schemes). The stability analysis is based on the appropriate a priori estimates for the intermediate problems. In this way, we have an opportunity to obtain estimates in finite-dimensional Banach spaces. Approaches to constructing additive schemes of componentwise splitting are investigated that have a higher (the second) order of accuracy; a relationship between these schemes and some other techniques, i.e., factorized schemes and methods of ADI type, is shown.

5.1 Additive formulations of differential problems We present some reasons explaining the fact that the solution of an unsteady problem can be determined numerically as a successive solution of simpler problems. We emphasize two classes of methods: schemes with the fractional and with the integer intermediate time steps.

5.1.1 Model problem Consider the Cauchy problem du C Au D f .t /, dt u.0/ D u0 .

t > 0,

(5.1) (5.2)

Assume, for simplicity, that the operator A in equation (5.1) is constant, i.e., it is independent of t . The operator of the problem is the p-component sum of constant operators: AD

p X

A˛ .

(5.3)

˛D1

The transition from the time level t n to the next one t nC1 will be based on the successive solution of the p Cauchy problems with the operators A˛ , ˛ D 1, 2, : : : , p.

145

Section 5.1 Additive formulations of differential problems

5.1.2 Intermediate problems Here we discuss the main approaches to the construction of additive schemes. In doing so, it is necessary to formulate auxiliary problems with operators A˛ , ˛ D 1, 2, : : : , p from the additive splitting (5.3); solving these problems, we obtain the approximate solution of the original problem (5.1), (5.2). For the construction of additive schemes, we employ two basic approaches: additive schemes with fractional time steps and additive schemes with integer steps. We introduce a uniform grid in time: !N  D ! [ ¹T º D ¹t n D n ,

n D 0, 1, : : : , N0 ,

N0 D T º.

Each half-interval .t n , t nC1  is split into p parts by means of the intermediate points t nC˛=p D t n C

˛ , p

˛ D 1, 2, : : : , p.

The equation (5.1), in view of the additive representation (5.3) for the operator A, may be written in the form  p  X 1 du (5.4) C A˛ u  f˛ .t / D 0, t > 0. p dt ˛D1 Here f˛ .t /, ˛ D 1, 2, : : : , p are arbitrary functions that satisfy the condition p X

f˛ .t / D f .t /,

t > 0.

(5.5)

˛D1

In accordance with (5.4), we solve the following sequence of intermediate problems. The auxiliary functions v˛ .t /, ˛ D 1, 2, : : : , p are determined from the equations 1 dv1 C A1 v1 D f1 .t /, t n < t  t nC1=p , p dt ::::::::::::::::::::::::::::::::::::::::::::::::::: 1 dv˛ C A˛ v˛ D f˛ .t /, p dt t nC.˛1/=p < t  t nC˛=p ,

˛ D 2, 3, : : : , p  1,

(5.6)

::::::::::::::::::::::::::::::::::::::::::::::::::: 1 dvp C A1 vp D fp , t nC.p1/=p < t  t nC1 . p dt The initial conditions for (5.6) have the form v1 .0/ D u0 ,

v1 .t n / D vp .t n /,

v˛ .t nC.˛1/=p / D v˛1 .t nC.˛1/=p /,

˛ D 2, 3, : : : , p.

(5.7)

146

Chapter 5 Schemes of summarized approximation

Here v.t n / D vp .t n / is treated as the approximate solution of (5.1)–(5.3) at the time moments t D t n . The intermediate problems (5.6), (5.7) are solved over individual parts of the halfinterval .t n , t nC1 . That is why the difference schemes based on (5.6), (5.7) are called additive fractional step schemes. The proximity of the solutions of (5.6), (5.7) and (5.1), (5.2) was considered by Yanenko in [216, 217]. It was shown that kv.t n /  u.t n /k D M  ,

M D const > 0

under the assumption of sufficient smoothness of the exact solution. The second method for constructing additive schemes (see [124]) is based on solving auxiliary problems over the entire time half-interval .t n , t nC1 . Therefore, such a scheme is referred to as the additive integer step scheme. To determine the functions v˛ .t /, ˛ D 1, 2, : : : , p, we employ the equations dv1 C A1 v1 D f1 .t /, t n < t  t nC1 , dt ::::::::::::::::::::::::::::::::::::::: dv˛ C A˛ v˛ D f˛ .t /, dt t n < t  t nC1 ,

˛ D 2, 3, : : : , p  1,

(5.8)

::::::::::::::::::::::::::::::::::::::: dvp C A1 vp D fp .t /, t n < t  t nC1 , dt supplemented with the initial conditions v1 .0/ D u0 ,

v1 .t n / D v .p/ .t n /,

v˛ .t n / D v˛1 .t nC1 /,

˛ D 2, 3, : : : , p.

(5.9)

In the numerical implementation of (5.8), (5.9), first, we solve the equation for v1 .t / with the initial condition v1 .t n / D vp .t n / and obtain v1 .t nC1 /, which is used as the initial value to evaluate v2 .t / and so on. The values v.t nC1 / D vp .t nC1 / are treated as the approximate solution of the problem (5.1)–(5.3) at the time level t nC1 . It should be noted that the two approaches considered here for constructing additive difference schemes, i.e., (5.6), (5.7) and (5.8), (5.9), have a close relationship. The solution of the system (5.8), (5.9) approximates the original problem (5.1)–(5.3) with the first order, too. If the operators A˛ , ˛ D 1, 2, : : : , p do not depend on time and the right-hand sides f˛ .t / D 0, ˛ D 1, 2, : : : , p, then the problems (5.6), (5.7) and (5.8), (5.9) are identical. In more general situations, it is preferable to use the second variant with solving auxiliary problems at integer time moments.

147

Section 5.1 Additive formulations of differential problems

5.1.3 Summarized approximation concept Now we study accuracy of the discrete analog for the system of equations (5.8), (5.9), i.e., the accuracy of the approximate solution of the problem (5.1)–(5.3). In doing so, we must extend the concept of approximation. Each individual intermediate problem (5.8), (5.9) does not approximate the original problem (does not yield an approximate solution), and only successively solving all intermediate problems with their correlation through the initial conditions allows us to obtain the approximate solution. That is why we say that the problem (5.8), (5.9) approximates (5.1)–(5.3) in a summarized sense (summarized approximation). Consider the error z1 .t / D v1 .t /  u.t /,

t n < t  t nC1 ,

z˛ .t / D v˛ .t /  u.t nC1 /,

˛ D 2, 3, : : : , p,

t n < t  t nC1 .

Let us formulate the corresponding problem for the error. Substituting z˛ .t /, ˛ D 1, 2, : : : , p into (5.8), (5.9), we obtain the equations dz˛ C A˛ z ˛ D dt n

t 0. dt

149

Section 5.1 Additive formulations of differential problems

The solution of (5.2), (5.13) at the time moment t D t nC1 is associated with the solution at the time moment t D t n by the relation u.t nC1 / D exp. .A1 C A2 //u.t n /. The application of the summarized approximation method (5.8), (5.9) corresponds to the representation v.t nC1 / D exp.A1 / exp.A2 /v.t n /.

(5.14)

For the error z.t n / D v.t n /  u.t n /, we have z.t nC1 / D exp.A1 / exp.A2 /z.t n / C 

n

,

where n

D .exp.A1 / exp.A2 /  exp. .A1 C A2 ///

un . 

Straightforward calculations give n

1 D  .A2 A1  A1 A2 /un C O. 2 /. 2

Therefore, for the commutative operators, the truncation error is of second order, but in the general case, it is of first order only. To improve the accuracy of the approximate solution of (5.2), (5.14), instead of (5.14), we employ the expression             v.t nC1 / D exp  A1 exp  A2 exp  A2 exp  A1 v.t n /. (5.15) 2 2 2 2 This corresponds to using, instead of (5.8), the following arrangement of calculations: dv˛ C A˛ v˛ D f˛ .t /, dt dv˛ C A˛ v˛ D f˛ .t /, dt

t n < t  t nC1=2 , t nC1=2 < t  t nC1 ,

˛ D 1, 2, : : : , p, ˛ D p, p  1, : : : , 1.

(5.16)

If we employ (5.16), then the truncation error is n D O. 2 / without assumptions about commutativity of the operators. The problem (5.8), (5.9) is attributed to the following chain: A1 7! A2 7!    Ap , which reflects the sequence of solving the intermediate problems. The symmetrized chain, associated with (5.16), has the form A1 7! A2 7!    Ap 7! Ap 7! Ap1 7!    A1 .

150

Chapter 5 Schemes of summarized approximation

The general case of multicomponent splitting is treated in a similar way. For (5.3)), we put p X Q Q A D A1 C A2 , A2 D A˛ . ˛D2

By (5.15), we obtain   v.t nC1 / D exp. A1 / exp. AQ2 / exp. A1 /v.t n /. 2 2 For the operator AQ2 in this representation, we apply a similar construction. As a result, we obtain p 1   Y    Y v.t nC1 / D exp  A˛ exp  A˛ v.t n /. 2 2 ˛D1 ˛Dp This leads to the above-mentioned arrangement (5.16) for solving the intermediate problems.

5.2 Investigation of schemes of summarized approximation Now we study additive schemes of summarized approximation with an arbitrary multicomponent splitting of a problem operator into pairwise noncommutative operators. We establish their stability, which is directly associated with the stability of the problems for individual components.

5.2.1 Schemes of componentwise splitting Let us consider the Cauchy problem du C Au D f .t /, dt u.0/ D u0 .

t > 0,

(5.17) (5.18)

Assume that the continuous operator A has the following general additive representation: p X A˛ , A˛  0, ˛ D 1, 2, : : : , p. (5.19) AD ˛D1

Here we do not assume that the operators A˛ , ˛ D 1, 2, : : : , p are pairwise commutative. Additive difference schemes for problems with splitting into three and more pairwise noncommutative operators are traditionally constructed using the concept of summarized approximation; in this case, we speak of schemes of componentwise splitting. If splitting is performed with respect to spatial variables, then these schemes are called locally one-dimensional schemes [62, 116].

151

Section 5.2 Investigation of schemes of summarized approximation

Schemes of multicomponent splitting correspond to a reduction of the original problem (5.17)–(5.19) to a chain of simpler problems (see (4.8), (4.9)), i.e., dv˛ C A˛ v˛ D f˛ .t /, dt v1 .0/ D u0 ,

t n < t  t nC1 ,

˛ D 1, 2, : : : , p,

(5.20)

v1 .t n / D vp .t n /,

v˛ .t n / D v˛1 .t nC1 /,

˛ D 2, 3, : : : , p.

(5.21)

Each intermediate problem is associate with a scheme with weights; this gives y nC˛=p  y nC.˛1/=p C A˛ .˛ y nC˛=p  C .1  ˛ /y nC.˛1/=p / D '˛n , ˛ D 1, 2, : : : , p,

(5.22) n D 0, 1, : : : .

In this scheme, the right-hand parts are consistent as follows: 'n D

p X

'˛n .

(5.23)

˛D1

Computations in the scheme (5.22) are arranged in such a way that the consistency conditions (5.21) are true for the solutions of the intermediate problems.

5.2.2 Estimates for the intermediate problem solutions For each individual equation in (5.20), we formulate the standard condition for stability. The necessary and sufficient condition for stability was formulated in Theorem 2.9. Here we present a levelwise estimate for stability with respect to the initial data and the right-hand side for the scheme with weights; it will be applied to analyze the additive scheme (5.22) that is based on multicomponent splitting. Consider the scheme with weights: y nC1  y n C A.y nC1 C .1   /y n / D ' n , 

t n 2 ! ,

(5.24)

where A  0, and the necessary and sufficient condition for stability   0.5 is valid. Our study is based on the investigation of the transition operator norm. Lemma 5.1. Suppose

C D  .E C  D/1 D,

(5.25)

where E is the identity operator and D  0. Then the following estimate holds: kE  C k  1,  > 0 for   =2.

(5.26)

152

Chapter 5 Schemes of summarized approximation

Proof. The inequality (5.26) is equivalent to the fulfillment of the operator inequality .E  C  /.E  C /  E, i.e., C  C  2C . Taking into account the specific representation (5.25) for the operator C , we have  .E C  D  /1 D  D.E C  D/1  2D.E C  D/1 . Multiplying this inequality by E C  D  from the left and by E C  D from the right (the inequality remains valid), we obtain D  D  2D C 2 D  D. It holds obviously for   =2. We can now return to the scheme with weights. Theorem 5.1. If   0.5 and  > 0, then the scheme .5.24/ satisfies the a priori estimate n X  k' k k. (5.27) ky nC1 k  ku0 k C kD0

Proof. Rewrite the scheme (5.24) in the form y nC1 D .E  C /y n C  .E C  A/1 ' n , where C D  .E C  A/1 A, i.e., in the representation (5.25), we have D D A. In view of Lemma 5.1 ( D 1) and we get ky nC1 k  kE  C kky n k C  k.E C  A/1 kk' n k  ky n k C  k' n k for   0.5. This implies the required estimate (5.27). Further a priori estimates for schemes with weights are given in the books [131,134, 136].

Section 5.2 Investigation of schemes of summarized approximation

153

5.2.3 Stability of componentwise splitting schemes Let us present a proper a priori estimate for stability (see [131, 134, 142]) with respect to the initial data and the right-hand side for the additive componentwise splitting scheme (5.19), (5.22). For the right-hand sides '˛n , ˛ D 1, 2, : : : , p in the scheme (5.23), we apply the special representation: '˛n

D

'N˛n

C

'Q˛n ,

˛ D 1, 2, : : : , p,

p X

'N˛n D 0.

(5.28)

˛D1

The following statement is true for the scheme of componentwise splitting. Theorem 5.2. Assume that 0.5  ˛  2, ˛ D 1, 2, : : : , p and  > 0. Then the scheme .5.19/, .5.22/ satisfies the a priori estimate ky

nC1

 p  p n X X X k k k  ku k C  'N˛ k . k'Q˛ k C  kA˛ 0

kD0

˛D1

(5.29)

ˇ D˛

Proof. For the approximate solution, we use the expression y nC˛=p D v nC˛=p C w nC˛=p ,

˛ D 1, 2, : : : , p,

(5.30)

where w 0 D 0. The functions w nC˛=p , ˛ D 1, 2, : : : , p are determined from the equations w nC˛=p  w nC.˛1/=p (5.31) D 'N˛n , ˛ D 1, 2, : : : , p.  In view of the decomposition (5.19) and our assumption about the initial value, this system of equations yields w nC1 D w n D    D w 0 D 0.

(5.32)

Consider the problem for the first term in (5.30). We have v nC˛=p  v nC.˛1/=p C A˛ .˛ v nC˛=p C .1  ˛ /v nC.˛1/=p /  D

n˛ ,

˛ D 1, 2, : : : , p,

n D 0, 1, : : : ,

where n˛ D 'Q˛n  A˛ w nC.˛1/=p  ˛ A˛

w nC˛=p  w nC.˛1/=p . 

By (5.31), it follows that w nC.˛1/=p D 

p X ˇ D˛

'Nˇn ,

˛ D 1, 2, : : : , p.

(5.33)

154

Chapter 5 Schemes of summarized approximation

Therefore, for the right-hand side of equation (5.33), we obtain p   X n˛ D 'Q˛n C A˛ .1  ˛ /'N˛n C 'Nˇn ,

˛ D 1, 2, : : : , p.

ˇ D˛C1

For the discrete equation (5.33) under the above restrictions that were formulated for ˛ , ˛ D 1, 2, : : : , p, we have the estimate p   X 'Nˇn k . kv nC˛=p k  kv nC.˛1/=p k C  k'Q˛n k C  kA˛ ˇ D˛

The summation over all ˛ from 1 up to p, in view of (5.32), yields the inequality ky nC1 k  ky n k C 

 p  p X X 'Nˇn k , k'Q˛n k C  kA˛

˛D1

ˇ D˛

which implies the desired estimate for stability with respect to the initial data and the right-hand side (5.29).

5.2.4 Convergence of componentwise splitting schemes The above stability estimate (5.29) provides the basis for studying the accuracy of the splitting schemes considered here. The problem for the error of the difference solution is formulated in the form (5.28), (5.29). The fundamental point for these componentwise splitting schemes is that stability estimates strongly depend on the splitting (5.19); more precisely, they depend on the individual terms appearing in the splitting. In fact, this means that the accuracy of such additive schemes depends on the formulation of intermediate problems. On the other hand, the intermediate problems (as well as the auxiliary grid functions y nC˛=p , y˛nC1 , ˛ D 1, 2, : : : , p) have no independent sense. Ideally, we would like to avoid their employment, i.e., we try to construct schemes without involving the concept of summarized approximation. Some opportunities in this promising direction are discussed in the following chapters of our book. Now we investigate the accuracy of the additive scheme of summarized approximation for the problem (5.17), (5.18). Let us consider the corresponding problem for the error. Assume that z n D y n  un and let z nC˛=p D y nC˛=p  unC˛=p , then the substitution of this notation into (5.22) yields z nC˛=p  z nC.˛1/=p C A˛ .˛ z nC˛=p C .1  ˛ /z nC.˛1/=p / D  ˛ D 1, 2, : : : , p, n D 0, 1, : : : .

n ˛,

(5.34)

155

Section 5.2 Investigation of schemes of summarized approximation

For the truncation error of the individual equations, we have n ˛

unC˛=p  unC.˛1/=p  A˛ .˛ unC˛=p C .1  ˛ /unC.˛1/=p / C '˛n ,  ˛ D 1, 2, : : : , p, n D 0, 1, : : : .

D

We use (see (5.28)) the representation of the truncation error in the form p X n n n N ˛n D 0. N Q D C , ˛ D 1, 2, : : : , p, ˛ ˛ ˛

(5.35)

˛D1

In view of (5.35), for the solutions of (5.17)–(5.19), we put N ˛n  1 du .t n /  A˛ u.t n / C f˛ .t n /, ˛ D 1, 2, : : : , p, p dt where, we recall, it is true that p X f˛ .t / D f .t /, t > 0.

(5.36)

˛D1

For the remaining terms of the truncation error, it follows from (5.34), (5.36) that Q ˛n D O. /,

˛ D 1, 2, : : : , p,

n D 0, 1, : : : .

(5.37)

If we specify the exact initial condition for (5.22), (5.23), then, in view of Theorem 5.2, the error of the approximate solution satisfies the estimate p p n X X X nC1 k N ˛k k/. Q kz k  .k ˛ k C  kA˛ kD0

˛D1

ˇ D˛

The substitution of (5.35)–(5.37) ensures the convergence of the additive componentwise splitting scheme (5.22), (5.23) with the first order in time.

5.2.5 Convergence of additive schemes in Banach spaces To study schemes of summarized approximation, we involve the appropriate stability estimate for the problems that are formulated for the individual operator terms. Such a scheme was already implemented (see equation (5.33)) for schemes of multicomponent splitting. So far, we focused on the corresponding estimates for stability in Hilbert spaces of grid functions, i.e., in H D L2 .!/. For many applied problems of mathematical physics, it is important to obtain estimates for stability and convergence in Banach spaces: in L1 .!/ and in L1 .!/. Let us consider, e.g., the componentwise splitting scheme (5.22), (5.23) for numerically solving the problem (5.17)–(5.19). Here we focus on the schemes where intermediate problems are stable in L1 .!/ or in L1 .!/. For these spaces, we have kwk1 D max jwi j, 1im

kAk1 D max

1im

m X j D1

jaij j,

156

Chapter 5 Schemes of summarized approximation

kwk1 D

m X

jwi j,

kAk1 D max

1j m

iD1

m X

jaij j

iD1

for a vector u D u.t / D ¹u1 , u2 , : : : , um º and a matrix A D ¹aij º. ˛ º ˛ D 1, 2, : : : , p are Assume that in the splitting (5.19), the matrices A˛ D ¹aij diagonally dominant by rows: ai˛i 

m X

˛ jaij j,

i D 1, 2, : : : , m,

(5.38)

i6Dj D1

or by columns: ˛ ajj



m X

˛ jaij j,

j D 1, 2, : : : , m,

˛ D 1, 2, : : : , p.

(5.39)

j 6DiD1

To derive unconditionally stable schemes, we take the weights in the additive scheme (5.22) such that ˛  1, ˛ D 1, 2, : : : , p. Then, if (5.38) (or (5.39)) is true, the following a priori estimate (see Theorem 2.17) holds: ky nC˛=p k  ky nC.˛1/=p k C  k'˛n k in the space L1 .!/ (or in L1 .!//. Repeating the proof of Theorem 5.2, we obtain the estimate (5.29). From this estimate under the above restrictions ˛  1, ˛ D 1, 2, : : : , p, we establish the unconditional convergence for the additive scheme (5.22), (5.23) with the first order by  in L1 .!/ or in L1 .!/.

5.3 Additively averaged schemes The above additive schemes of summarized approximation are based on the successive solution of p simpler problems, where the solution of one problem provides the initial conditions for the next one. The second possibility, discussed below, allows to construct additive schemes with independent solving of simpler problems via parallel algorithms.

5.3.1 Differential problem We illustrate the idea of constructing a new class of additive schemes using a model problem with the two-component splitting du C .A1 C A2 /u D 0, dt u.0/ D u0 .

t > 0,

(5.40) (5.41)

157

Section 5.3 Additively averaged schemes

The exact solution of the problem (5.40), (5.41) at a time level is written as u.t nC1 / D exp. .A1 C A2 //u.t n /. Now we define the approximate solution by the equation 1 v.t nC1 / D .exp.2A1 / C exp.2A2 //v.t n /. (5.42) 2 If we use the usual scheme of componentwise splitting, then a differential analog has the form v.t nC1 / D exp.A1 / exp.A2 /v.t n /. In this case, we have a multiplicative version of splitting schemes (the operator exponentials are multiplied), whereas (5.42) is attributed to the additive variant (the operator exponentials are added). The problem for the error z.t n / D v.t n /  u.t n / has the form 1 z.t nC1 / D .exp.2A1 / C exp.2A2 //z.t n / C  2

n

,

where n

D

1 .exp.2A1 / C exp.2A2 // 2 1  exp. .A1 C A2 //un . 

We get n D O. /, i.e., the truncation error is of the first order. In the case of commutative operators, the truncation error is also of the first order. The multicomponent case is treated in a similar way. For the splitting AD

p X

A˛ ,

A˛  0,

˛ D 1, 2, : : : , p,

(5.43)

˛D1

the transition from one time level to the next one is carried out as v.t nC1 / D

p 1 X exp.pA˛ /v.t n /. p ˛D1

(5.44)

In fact, the solution is the arithmetic mean for the problems associated with the individual operators A˛ , ˛ D 1, 2, : : : , p.

5.3.2 Additive schemes Let us construct difference schemes to solve numerically the Cauchy problem for the equation du C Au D f .t /, t > 0 (5.45) dt

158

Chapter 5 Schemes of summarized approximation

with the additive operator (5.43) and the right-hand side p X

f˛ .t / D f .t /,

t > 0.

˛D1

For this problem, computations through (5.44) correspond to solving the problems 1 dv˛ C A˛ v˛ D f˛ .t /, p dt

t n < t  t nC1 ,

˛ D 1, 2, : : : , p

(5.46)

with the initial conditions v˛ .t n / D v.t n /

(5.47)

and evaluating the solution at t D t n in accordance with v.t nC1 / D

p 1 X v˛ .t nC1 /. p ˛D1

(5.48)

Thus, we solve independently p problems (5.46), (5.47) with the same initial conditions (5.47), and then the arithmetic mean (5.48) is treated as the approximate solution at the new time level. Additive schemes of this class were proposed in the paper [50] and are called additively averaged schemes of componentwise splitting. The abovementioned possibility of asynchronous (independent) computations is particularly important in constructing numerical algorithms for parallel computers. In accordance with (5.46)–(5.48), the transition to the new time level is conducted as follows: y˛nC1  y n C A˛ .˛ y˛nC1 C .1  ˛ /y n / p D '˛n , ˛ D 1, 2, : : : , p, n D 0, 1, : : : , y nC1

(5.49)

p 1 X nC1 D y . p ˛D1 ˛

Here the representation 'n D

p X

'˛n .

(5.50)

˛D1

is used for the right-hand side.

5.3.3 Stability of additively averaged schemes Conditions for stability of additively averaged schemes are the same as those for the standard componentwise splitting schemes. Similarly to Theorem 5.2, we prove the following main result.

159

Section 5.3 Additively averaged schemes

Theorem 5.3. For ˛  0.5, ˛ D 1, 2, : : : , p and any  > 0, the solution of .5.43/, .5.49/, .5.50/ satisfies the a priori estimate ky

nC1

p n X X k  ku k C  .k'Q˛k k C p ˛ kA˛ 'N˛k k/. 0

(5.51)

˛D1

kD0

Proof. Again we apply the representation y˛nC1 D v˛nC1 C w˛nC1 ,

˛ D 1, 2, : : : , p

(5.52)

with w0 D 0. To find functions w˛nC1 , ˛ D 1, 2, : : : , p, we use the equations w˛nC1  w n D 'N˛n , p By (5.50), for the mean quantities w

nC1

˛ D 1, 2, : : : , p.

(5.53)

p 1 X nC1 D w p ˛D1 ˛

from (5.53), we get w nC1 D w n D : : : D w0 D 0. From (5.49), (5.52), we have v˛nC1  v n C A˛ .˛ v˛nC1 C .1  ˛ /v n / D n˛ , p ˛ D 1, 2, : : : , p, n D 0, 1, : : : , v nC1 D

1 p

p X

(5.54)

v˛nC1 ,

˛D1

where n˛ D 'Q˛n  p˛ A˛

w˛nC1  w n . p

In view of (5.53), we derive n˛ D 'Q˛n  p ˛ A˛ 'N˛n ,

˛ D 1, 2, : : : , p.

For the discrete equation (5.54), by Theorem 4.1, we have kv nC˛=p k  kv nC.˛1/=p k C p .k'Q˛n k C p kA˛ 'N˛n k/. The summation over all ˛ from 1 up to p leads to the inequality ky nC1 k  ky n k C 

p X

.k'Q˛n k C  kA˛

˛D1

p X

'Nˇn k/,

ˇ D˛

which implies the desired estimate for stability with respect to the initial data and the right-hand side (5.51).

160

Chapter 5 Schemes of summarized approximation

Based on this estimate, we establish in the standard way that the additively averaged componentwise splitting scheme (5.50) converges with the first order by  . A potential advantage of the additively averaged scheme (5.49) is that it allows us to apply a clear parallel implementation for computing grid functions y˛nC1 , ˛ D 1, 2, : : : , p.

5.4 Other variants of componentwise splitting schemes In particular cases, additive schemes with componentwise splitting may be associated with factorized schemes. The classical ADI schemes with a special arrangement of computations may be treated as additive schemes. Special attention should be given to issues of constructing and studying additive schemes with second-order accuracy.

5.4.1 Fully implicit additive schemes Let us consider an important special case of the additive scheme with componentwise splitting, where, in (4.27), we have the weight parameters ˛ D 1, ˛ D 1, 2, : : : , p, i.e., y nC˛=p  y nC.˛1/=p C A˛ y nC˛=p D '˛n , 

˛ D 1, 2, : : : , p.

(5.55)

For the right-hand side, we put '1n D ' n ,

'˛n D 0,

˛ D 2, 3, : : : , p.

(5.56)

From (5.55), (5.56), we get the recurrence relations .E C A1 /y nC1=p D y n C  ' n , .E C A˛ /y nC˛=p D y nC.˛1/=p ,

˛ D 2, 3, : : : , p,

t n 2 ! .

The elimination of intermediate solutions leads to the scheme p Y

.E C A˛ /y nC1 D y n C ' n ,

t n 2 ! .

(5.57)

˛D1

Thus, we have a special factorized scheme, which was previously constructed (see (3.81)) using the regularization principle for difference schemes. The scheme (5.57) approximates the differential problem with the first order by  . The stability and convergence of the componentwise splitting scheme (5.55), (5.56) follows from a general consideration of additive componentwise splitting schemes.

Section 5.4 Other variants of componentwise splitting schemes

161

5.4.2 ADI methods as additive schemes We will show that the classical ADI methods may be interpreted as a special additive scheme of summarized approximation. Let us consider the problem du C Au D f .t /, t > 0, (5.58) dt u.0/ D u0

(5.59)

with the two-component splitting A D A1 C A2 ,

A˛  0,

˛ D 1, 2.

(5.60)

In the classical Peaceman–Rachford algorithm, the approximate solution of the problem (5.58)–(5.60) is determined from y nC1=2  y n C A1 y nC1=2 C A2 y n D ' n , 0.5

(5.61)

y nC1  y nC1=2 (5.62) C A1 y nC1=2 C A2 y nC1 D ' n . 0.5 Write (5.61), (5.62) as an equivalent additive scheme of componentwise splitting. We introduce the four-component splitting for the operator of the problem (5.58), (5.59), i.e., 4 X AQ˛ , AQ˛  0, ˛ D 1, 2, : : : , 4, AD ˛D1

where

1 1 1 1 AQ1 D A2 , AQ2 D A1 , AQ3 D A1 , AQ4 D A2 . 2 2 2 2 A similar representation is also used for the right-hand side 4 X 'n D '˛n , ˛D1

where

1 1 '1n D ' n , '2n D 0, '3n D 0, '4n D ' n . 2 2 Let us apply a special variant of the componentwise splitting scheme that is characterized by a particular choice of weights: 1 1 y nC1=4  y n C A2 y n D ' n ,  2 2 y nC1=2  y nC1=4 1 C A1 y nC1=2 D 0,  2 y nC3=4  y nC1=2 1 C A1 y nC1=2 D 0,  2 y nC1  y nC3=4 1 1 C A2 y nC1 D ' n ,  2 2

n D 0, 1, : : :

162

Chapter 5 Schemes of summarized approximation

Eliminating y nC1=4 , y nC3=4 , we get     E C A1 y nC1=2 D E   2   E C A2 y nC1 D E  2

  n A2 y C  ' n , 2   A1 y nC1=2 C  ' n . 2

We can rewrite the Peaceman–Rachford method ((5.61), (5.62)) just in this form. Direct calculations indicate that the truncation error seems as follows: n

D

4 X

n ˛

D O. 2 /,

˛D1

i.e., the scheme has the summarized approximation O. 2 /.

5.4.3 Additive schemes with second-order accuracy The standard additive schemes of componentwise splitting have first-order accuracy by  . To improve their accuracy, we use symmetrization, i.e., the transition from the sequence of problems: A1 7! A2 7!    Ap , to the chain A1 7! A2 7!    Ap 7! Ap 7! Ap1 7!    A1 . In fact, such a scheme is implemented in the additive schemes above, which are equivalent to ADI methods. Additive schemes of componentwise splitting with second-order accuracy are studied in many works (see, e.g., [38, 39]). To construct additive schemes of componentwise splitting with second-order accuracy, we introduce the intermediate problems over the half-interval .t n , t nC1 : 1 dv˛ C A˛ v˛ D f˛ .t /, dt 2 1 dv˛ C A˛ v˛ D f˛ .t /, dt 2 t n < t  t nC1 ,

˛ D 1, 2, : : : , p, (5.63)

˛ D p, p  1, : : : , 1

with the right-hand sides 2p X ˛D1

f˛ .t / D f .t /,

t > 0.

163

Section 5.4 Other variants of componentwise splitting schemes

We associate the system of equations (5.63) with the simplest additive scheme y nC˛=2p  y nC.˛1/=2p 1 C A˛ .y nC˛=2p C y nC.˛1/=2p /  2 D '˛n ,

˛ D 1, 2, : : : , p,

1 y nC˛=2p  y nC.˛1/=2p C A2pC1˛ .y nC˛=p C y nC.˛1/=p /  2 D '˛n ,

(5.64)

˛ D p C 1, p C 2, : : : , 2p,

n D 0, 1, : : : . The stability of the additive scheme follows from general conditions for stability (see Theorem 5.2). The proof of convergence with the second order by  requires a special investigation.

5.4.4 Convergence of higher-order schemes Let us consider the additive scheme (5.64) with the right-hand sides specified as 1 f .t nC1=2 /. '˛n D 2p To solve numerically the problem (5.58), (5.59), we define the error z n D y n  un and z nC˛=2p D y nC˛=2p  unC˛=2p , ˛ D 1, 2, : : : , 2p. From (5.64), it follows that z nC˛=2p  z nC.˛1/=2p 1 C A˛ .z nC˛=2p C z nC.˛1/=2p /  2 D

n ˛,

˛ D 1, 2, : : : , p,

1 z nC˛=2p  z nC.˛1/=2p C A2pC1˛ .z nC˛=2p C z nC.˛1/=2p /  2 D

n ˛,

(5.65)

˛ D p C 1, p C 2, : : : , 2p,

n D 0, 1, : : : For the truncation error of individual equations, we get unC˛=2p  unC.˛1/=2p 1  A˛ .unC˛=2p C unC.˛1/=2p / C '˛n ,  2 ˛ D 1, 2, : : : , p,

n ˛

D

n ˛

D

unC˛=2p  unC.˛1/=2p   ˛ D p C 1, p C 2, : : : , 2p,

1 A2pC1˛ .unC˛=2p C unC.˛1/=2p / C '˛n , 2 n D 0, 1, : : : .

164

Chapter 5 Schemes of summarized approximation

Thus, we have unC˛=2p  unC.˛1/=2p 1  A˛ .unC˛=2p C unC.˛1/=2p / C '˛n  2 1 du nC.˛1=2/=2p 1 1 D /  A˛ u.t nC.˛1=2/=2p / C .t f .t nC1=2 / C O. 2 /, 2p dt 2 2p ˛ D 1, 2, : : : , p. With this in mind, for the truncation error, we use the representation n ˛

D N ˛n C O ˛n C Q ˛n ,

˛ D 1, 2, : : : , 2p.

(5.66)

For simplicity, assume that the operators A˛ , ˛ D 1, 2, : : : , p are constant, then N ˛n D  1 du .t nC1=2 /  1 A˛ u.t nC1=2 / C 1 f .t nC1=2 / D O.1/, 2p dt 2 2p  2  d 1 u du O ˛n D .t nC.˛1=2/=2p  t nC1=2 / C A˛ .t nC1=2 / dt 2 2 dt D O. /, ˛ D 1, 2, : : : , p, N ˛n D  1 du .t nC1=2 /  1 A.2pC1˛/ u.t nC1=2 / C 1 f .t nC1=2 / D O.1/, 2p dt 2 2p  2  O ˛n D .t nC.˛1=2/=2p  t nC1=2 / d u C 1 A.2pC1˛/ du .t nC1=2 / D O. /, dt 2 2 dt ˛ D p C 1, p C 2, : : : , 2p, n Q ˛ D O. 2 /, ˛ D 1, 2, : : : , 2p. Further investigation is conducted similarly to the proof of Theorem 5.2. We apply the representation z nC˛=2p D q nC˛=2p C w nC˛=2p ,

˛ D 1, 2, : : : , 2p

with w 0 D 0. The functions w nC˛=2p , ˛ D 1, 2, : : : , 2p satisfy the equations w nC˛=2p  w nC.˛1/=2p D N ˛n C O ˛n ,  Taking into account that

2p  X

˛ D 1, 2, : : : , 2p.

 N ˛n C O ˛n D 0,

˛D1

we get w nC1 D w n D    D w 0 D 0.

(5.67)

Section 5.4 Other variants of componentwise splitting schemes

165

For the functions q nC˛=2p , ˛ D 1, 2, : : : , 2p, we obtain the equations 1 q nC˛=2p  q nC.˛1/=2p C A˛ .q nC˛=2p C v nC.˛1/=2p / D n˛ ,  2 ˛ D 1, 2, : : : , p, 1 q nC˛=2p  q nC.˛1/=2p C A2pC1˛ .q nC˛=2p C v nC.˛1/=2p / D n˛ ,  2 ˛ D p C 1, p C 2, : : : , 2p, where 1 w nC˛=2p  w nC.˛1/=2p , n˛ D 'Q˛n  A˛ w nC.˛1/=2p  A˛ 2  ˛ D 1, 2, : : : , p, 1 w nC˛=2p  w nC.˛1/=2p n˛ D 'Q˛n  A2pC1˛ w nC.˛1/=2p  A2pC1˛ , 2  ˛ D p C 1, p C 2, : : : , 2p. The solution of equation (5.67) has the form p p X X nC.˛1/=2p n D 'Nˇ C  'Oˇn , w ˇ D˛

˛ D 1, 2, : : : , 2p.

ˇ D˛

Now we write the right-hand sides of the equations for q nC˛=2p , ˛ D 1, 2, : : : , 2p in the form n˛ D N n˛ C Q n˛ , ˛ D 1, 2, : : : , 2p, where



Q n˛

D

'Q˛n

C A˛

 2p X 1 n n 'Oˇ , 'O C 2 ˛ 

Q n˛ D 'Q˛n C A2pC1˛

˛ D 1, 2, : : : , p,

ˇ D˛C1

 2p X 1 n 'Oˇn , 'O˛ C 2

˛ D p C 1, p C 2, : : : , 2p,

ˇ D˛C1

O. 2 /,

D ˛ D 1, 2, : : : , 2p. i.e., For the terms N n˛ , ˛ D 1, 2, : : : , 2p, we have   2p X 1 n n n N ˛ D A˛ 'N˛ C 'Nˇ , ˛ D 1, 2, : : : , p, 2 Q n˛

 N n˛

D A

.2pC1˛/

ˇ D˛C1

 2p X 1 n n 'Nˇ , 'N C 2 ˛ ˇ D˛C1

˛ D p C 1, p C 2, : : : , 2p.

166

Chapter 5 Schemes of summarized approximation

Straightforward calculations yield the equality N n˛ D N n2pC1˛ , i.e.,

2p X

˛ D 1, 2, : : : , p,

N n˛ D 0.

˛D1

D O. /, ˛ D 1, 2, : : : , 2p, we establish the levelUsing Theorem 5.2, in view of wise estimate kq nC1 k  kq n k C O. 2 /. N n˛

A similar estimate holds for the error z nC1 , too. Thus, the symmetrized additive scheme of componentwise splitting (5.64) converges with the second order with respect to  .

Chapter 6

Regularized additive schemes

The regularization principle for difference schemes provides great opportunities in constructing difference schemes of a prescribed quality. Using this principle, stable difference schemes have been constructed for a wide class of mathematical physics problems as well as for iterative methods to solve difference equations. In this part of the book, we discuss some new results on designing stable difference schemes via perturbations of scheme operators. Considering the Cauchy problem for an evolutionary equation of the first order, we study the additive and multiplicative regularization for constructing unconditionally stable difference schemes. A new class of additive schemes (splitting schemes) of full approximation is proposed for arbitrary multicomponent splitting. Possibilities of constructing regularized additive schemes of secondorder accuracy are analyzed, and regularized additive difference schemes for secondorder equations are studied.

6.1 Multiplicative regularization of difference schemes The standard approach to the construction of stable schemes on the basis of the regularization principle is associated with the introduction of additional terms (regularizers) in operators of a generating difference scheme; in this case, we speak of additive regularization. Here we emphasize a class of multiplicative regularization methods where scheme operators are perturbed by means of operator multipliers.

6.1.1 Regularization principle for difference schemes The regularization theory for difference schemes is considered as the principle of improving the quality of a scheme by introducing regularizers into the operators of the original difference scheme. Unconditionally stable schemes are constructed on the basis on the regularization principle as follows: 1. A simple scheme, which is called a generating scheme, is constructed for the given problem; this scheme does not have the desired properties, i.e., it may be conditionally stable or even absolutely unstable. 2. This scheme is written in a unified (canonical) form for which the stability conditions are known. 3. The scheme quality (its stability) is improved by modifying its operators.

168

Chapter 6 Regularized additive schemes

Thus, the regularization principle for difference schemes is based on the known results concerning stability conditions. Stability criteria are given by the general stability theory for difference schemes [131, 134]. From this viewpoint, we can consider the regularization principle as a means of the constructive application of the general results of the stability theory for difference schemes. This is achieved by rewriting a scheme in a fairly general canonical form and by the formulation of stability criteria in a form that is convenient for checking. Let H be a finite-dimensional Hilbert space, and let D and A be linear operators in H . The scalar product and the norm in H are denoted by ., / and k  k, respectively. For D D D  > 0, we denote by HD the space H equipped with the scalar product and the norm .y, v/D D .Dy, v/, kyk D .Dy, y/1=2 . Let  > 0 be a time step, and y n D y.t n /, t n D n . Consider a homogeneous (with the zero right-hand side) two-level operator-difference scheme in the canonical form B

y nC1  y n C Ay n D 0, 

n D 0, 1, : : :

(6.1)

for a given y 0 . We assume that the operators A and B in (6.1) are constant (independent of n) and A is self-adjoint and positive (A D A > 0). We recall the main results (see Theorems 2.2 and 2.3) on stability of two-level difference schemes with respect to the initial data. For the scheme (6.1) with the operator A D A > 0, the condition B

 A 2

(6.2)

is necessary and sufficient for stability in HA , i.e., for the validity of the estimate ky nC1 kA  ky n kA ,

n D 0, 1, : : :

(6.3)

If the operator B D B  > 0, then the condition (6.2) is also necessary and sufficient for the stability in HB .

6.1.2 Additive regularization Let us consider the model Cauchy problem du C Au D 0, dt

t > 0,

u.0/ D u0 with a linear operator A in H that is constant, self-adjoint, and positive.

(6.4) (6.5)

169

Section 6.1 Multiplicative regularization of difference schemes

According to the regularization principle, first, we take a difference scheme that will be used as the starting point. It is natural to use the simple explicit scheme y nC1  y n C Ay n D 0, 

n D 0, 1, : : : ,

(6.6)

y 0 D u0

(6.7)

as such a generating scheme. Write the scheme (6.6) in the canonical form for two-level operator-difference schemes, i.e., in the form (6.1), with the operator B D E, where E is the identity operator. Taking into account the inequality A  kAkE from the necessary and sufficient stability condition (6.2) we derive the following restriction on the time step for the explicit scheme (6.6), (6.7): 2 .  kAk According to (6.2), stability can be improved in two ways. The first one improves stability by increasing the energy .By, y/ of the operator B (the left-hand side of the inequality (6.2)); the other method is to decrease the energy of the operator A (the right-hand side of the inequality (6.2)). First, we consider the possibilities opened by adding operator terms to the operators B and A. In this case, we deal with additive regularization. It is most natural to begin with an additive perturbation of the operator B, i.e., to make the change B 7! B C R, where R is a regularizing operator and  stands for a regularization parameter. Taking into account the fact that B D E in the generating scheme, we set B D E C R. (6.8) To retain the first-order approximation with respect to time in the scheme (6.1), (6.8), it is sufficient to set  D O. /. We consider two typical variants of the regularization operator: R D A,

(6.9)

RDA .

(6.10)

2

Theorem 6.1. The regularized scheme .6.1/, .6.8/ is stable in HA if ˛  2 .6.9/ and if ˛  16 in the case .6.10/.

 2

in the case

The regularized scheme (6.1), (6.8), (6.9) corresponds to the use of the standard scheme with weights y nC1  y n C A.y nC1 C .1   /y n / D 0,  with  D   .

n D 0, 1, : : :

170

Chapter 6 Regularized additive schemes

There is another possibility of additive regularization by perturbing the operator A, where the change A 7! AR is made. An example is the choice of the regularizing operator by the rule (6.10) (see, e.g., [159]). We emphasize that the scheme remains explicit in this case. For the scheme B

y nC1  y n Q n D 0, C Ay 

n D 0, 1, : : :

where

AQ D A  A2 , B D E, the non-negativity of the operator AQ and the stability condition lead to the following restrictions on the regularization parameter:  1 ,  .  kAk 8 8 , i.e., we can increase the maximal kAk admissible time step for the explicit scheme by a factor of four. For the optimal choice of , we have  

6.1.3 Multiplicative regularization The conventional approach to the construction of stable schemes is based on the use of additive regularization. Another possibility is to employ a multiplicative perturbation of discrete operators of the generating scheme. Let us discuss simple examples of using this approach; some of them may be treated as a new interpretation of the regularized operator-difference schemes examined above. The multiplicative regularization of the operator B assumes that the change B 7! B.1 C R/ or B 7! .1 C R/B is made. In this case, we remain in the class of schemes with self-adjoint operators if RB D BR . Then we have the regularized scheme (6.1), (6.8) discussed above. An example of a more complicated regularization is given by the transformation B 7! .E C R /B.E C R/.  In the case where R D A, the stability condition has the form   . Another inter8 esting example of such a regularization corresponds to the alternating triangle method,   where A D R C R and   . 2 Similarly, we can regularize the scheme by a multiplicative perturbation of the operator A. In view of the inequality (6.2), we can perform the transformation A 7! A.1 C R/1 or A 7! .1 C R/1 A. For simple two-level schemes, such a regularization may be treated as a version of the regularization of the operator B. To remain in the class of schemes with self-adjoint operators, it is sufficient to set R D R.A/. Great opportunities are connected with the regularization A 7! .E C R /1 A.E C R/1 .

Section 6.2 Multiplicative regularization of additive schemes

171

In this case, the regularizing operator R cannot be directly associated with the operator A.

6.2 Multiplicative regularization of additive schemes We construct additive difference schemes for evolutionary equations of first order considering the general case of additive splitting with an arbitrary number of noncommutative operator pairs. The construction of unconditionally stable schemes is based on the regularization of a simple explicit two-level scheme by means of small multiplicative perturbations of each operator term in the splitting.

6.2.1 The Cauchy problem for a first-order equation Additive schemes are characterized by a decomposition (splitting) of a problem operator into the sum of operators with a simpler structure. In the general case of the splitting of a problem operator into the sum of noncommutative non-self-adjoint terms, additive difference schemes are most easily constructed for the two-component splitting. A more complicated situation occurs in the case of a multicomponent splitting (into three and more operators). For such problems, the most interesting results were obtained using the concept of summarized approximation. The original problem in the transition from one time level to another is divided into several subproblems, and each of these subproblems does not approximate, in general, the original problem. Unconditionally stable componentwise splitting schemes (locally one-dimensional schemes of splitting with respect to spatial variables) are constructed in this way. If we focus on modern parallel computers, then special attention should be given to additively averaged schemes of componentwise splitting. A new class of operator-difference splitting schemes – vector additive schemes of full approximation – is developed for general multicomponent splitting. In the above-discussed additive schemes, for the transition to a new time level, we introduced several intermediate (auxiliary) functions, which did not always have a direct relation to the desired solution. Consider grid functions y in a real finite-dimensional Hilbert space H equipped with the scalar product and the norm .y, w/, kyk D .y, y/1=2 , respectively. For D D D  > 0, we denote by HD the space H equipped with the scalar product .y, w/D D .Dy, w/ and the norm kykD D .Dy, y/1=2 . In the Cauchy problem for the first-order evolutionary equation, we seek a function y.t / 2 H satisfying the equation du C Au D f .t /, dt

t >0

(6.11)

and the initial condition u.0/ D u0 .

(6.12)

172

Chapter 6 Regularized additive schemes

Assume that the linear operator A acting from H onto H (A : H ! H ) is positive and time-independent (A > 0). The solution of the problem (6.11), (6.12) satisfies the a priori estimate Z t kf .s/kds, (6.13) ku.t /k  ku0 k C 0

which reflects stability with respect to the initial data and the right-hand side. Assume that the operator A may be represented in the additive form AD

p X

A˛ ,

A˛  0,

˛ D 1, 2, : : : , p.

(6.14)

˛D1

Additive schemes are constructed using the decomposition (6.14), and the transition from the time level t n to the next level t nC1 D t n C involves the solution of problems for the individual operators A˛ , ˛ D 1, 2, : : : , p in the additive decomposition (6.14), i.e., the problem is split into p subproblems.

6.2.2 Regularization of additive schemes To construct difference schemes with a prescribed quality, we can apply multiplicative regularization. Considering the problem (6.11), (6.12), it is natural to take the explicit scheme y nC1  y n (6.15) C Ay n D ' n , n D 0, 1, : : :  with a given y 0 D u0 as the generating scheme. The scheme (6.15) is written in the canonical form y nC1  y n Q n D 'Q n , n D 0, 1, : : : (6.16) C Ay BQ  where BQ D A1 , AQ D E. The necessary and sufficient condition for stability of the scheme (6.16) in H is formulated in the form of the inequality  Q (6.17) BQ  A. 2 In accordance with (6.17), to improve the stability condition, we focus on a perturbaQ or on a perturbation (decreasing) of the operator A. Q tion (increasing) of the operator B, The multiplicative regularization of the explicit scheme (6.15) yields y nC1  y n C .E C A/1 Ay n D ' n , n D 0, 1, : : :  This scheme for  D   differs from the conventional scheme with weights y nC1  y n C A.y nC1 C .1   /y n / D ' n ,  only in the right-hand side.

n D 0, 1, : : :

(6.18)

(6.19)

Section 6.2 Multiplicative regularization of additive schemes

173

The schemes (6.18) and (6.19) with   0.5 (  0.5 ) satisfy the a priori estimate ky

nC1

k  ku k C 0

n X

 k'k k,

(6.20)

kD0

which guarantees stability in H and, moreover, it is consistent with the estimate (6.13) for the original problem (6.11), (6.12). The principal feature of the scheme (6.18) is connected with the fact that it is based on the explicit scheme with multiplicative regularization of the problem operator. In this way, we can also design additive schemes. To construct unconditionally stable additive schemes for the problem (6.11), (6.12), (6.14), we use as the generating scheme the simple explicit scheme y nC1  y n X A˛ y n D ' n , C  ˛D1 p

n D 0, 1, : : :

(6.21)

By analogy with (6.18), we construct additive schemes via the perturbation of each individual term in the operator of the scheme (6.21): y nC1  y n X .E C A˛ /1 A˛ y n D ' n , n D 0, 1, : : : C  ˛D1 p

(6.22)

In the regularized additive scheme [146, 150], we have no intermediate problem (auxiliary functions are not introduced); the original problem is directly approximated. This is a fundamental feature of this class of additive schemes. Restrictions on a time step for the explicit scheme are associated with the norm of the problem operator. The above multiplicative regularization is aimed at decreasing this norm in an appropriate way.

6.2.3 Stability and convergence Regularized additive schemes in their form (but not in their implementation) are explicit schemes. Based on standard conditions for stability of explicit schemes, we also formulate conditions for the stability of the regularized additive scheme. Theorem 6.2. For   p=2 and any  > 0, the scheme .6.22/ satisfies the a priori estimate .6.20/. Proof. Rewrite (6.22) in the form y nC1  y n Q n D ' n, C Ay 

n D 0, 1, : : : ,

(6.23)

174

Chapter 6 Regularized additive schemes

where AQ D

p X

AQ˛ ,

AQ˛ D .E C A˛ /1 A˛ ,

˛D1

(6.24)

˛ D 1, 2, : : : , p. The inequalities D  0,

.E C A˛ /D.E C .A˛ / /  0,

˛ D 1, 2, : : : , p

are equivalent, and therefore if A˛  0, ˛ D 1, 2, : : : , p, then the operators AQ˛  0, ˛ D 1, 2, : : : , p are also non-negative. Thus, in the problem (6.23), (6.24), we have AQ  0. For the solution at the new time level, we have Q n C  ' n, y nC1 D .E   A/y and so

n Q k C  k' n k. ky nC1 k  kE   Akky

By (6.24), we get Q  kE   Ak

p 1 X kE  p AQ˛ k. p ˛D1

We show that the above constraints on the regularization parameter  ensures the estimate (6.25) kE  p AQ˛ k  1 for each operator AQ˛ , ˛ D 1, 2, : : : , p. The inequality (6.25) is equivalent to .E  p AQ˛ /.E  p AQ˛ /  E, i.e.,

p AQ˛ AQ˛  AQ˛ C AQ˛ .

To prove this fact, we multiply the inequality from the left by E C A˛ and from the right by E C A˛ (the inequality does not change). In view of (6.24), it gives pA˛ A˛  A˛ C A˛ C .A˛ A˛ C A˛ A˛ /. The inequality holds for   0.5p . In view of (6.25), the approximate solution satisfies the levelwise estimate ky nC1 k D ky n k C  k' n k, and therefore it follows that (6.20) holds.

Section 6.2 Multiplicative regularization of additive schemes

175

Using the a priori estimate (6.20), we can study in the usual manner the accuracy of the additive scheme (6.22). It is easy to prove that it converges with the first order in time. Consider the corresponding problem for the error z n D y n  un . For  D   , we have z nC1  z n Q n D n , n D 0, 1, : : : , (6.26) C Az  where the truncation error unC1  un Q n C 'n,  Au  Taking into account (6.24), we obtain n

D

AQ˛ D A˛    .A˛ /2 C O. 2 /,

n D 0, 1, : : :

˛ D 1, 2, : : : , p.

For the truncation error, this yields d 2u n D  2 .t n / C   .A˛ /2 C O. 2 /. dt The solution of (6.26), in view of Theorem 6.2, satisfies the estimate n X kz nC1 k   k'k k, kD0

which ensures the convergence of the regularized additive scheme (6.23), (6.24) for  D   ,   0.5p with the first order by  .

6.2.4 Regularized and additively averaged schemes The regularized scheme (6.22) has a close relationship with the additively averaged scheme of summarized approximation. To demonstrate this fact, we introduce a fictitious grid values y˛nC1 , ˛ D 1, 2, : : : , p. These functions have no independent sense. We implement the scheme (6.22) in the following form: y˛nC1  y n 1 C A˛ y n D .E C  A˛ /' n , p p ˛ D 1, 2, : : : , p, n D 0, 1, : : : .

.E C  A˛ /

The solution at the new time level is determined by the formula y

nC1

p 1 X nC1 D y . p ˛D1 ˛

Thus, we come to the additively averaged scheme, which is constructed without involving the concept of summarized approximation. This scheme differs from the standard additively averaged schemes of componentwise splitting in the choice of the right-hand sides. We emphasize that this is only one of the possible realizations of regularized difference schemes.

176

Chapter 6 Regularized additive schemes

6.3 Schemes of higher-order accuracy Here we construct regularized additive schemes of the second-order approximation by  in order to solve numerically the Cauchy problem for an evolutionary equation of first order. The three-level explicit scheme is taken as the generating scheme.

6.3.1 Statement of the problem The construction of additive difference schemes of the second order is conducted under stronger conditions. Namely, in addition, we require self-adjointness of a splitting operator. Let us consider a model problem du C Au D f .t /, t > 0, dt u.0/ D u0 , where AD

p X

A˛ > 0,

A˛ D A˛  0,

˛ D 1, 2, : : : , p

(6.27) (6.28)

(6.29)

˛D1

and all operators A˛ , ˛ D 1, 2, : : : , p are constant. Designing additive schemes under more general conditions requires a special investigation. We highlight classes of the standard difference schemes with the second-order approximation for the problem (6.27), (6.28) that provide the basis for constructing additive difference schemes. The simplest scheme of this type is the symmetric two-level scheme (the Crank–Nicolson scheme) y nC1 C y n y nC1  y n CA D 'n,  2

n D 0, 1, : : : ,

where ' n D f .t nC1=2 /. Three-level schemes provide many more possibilities. The most famous example is the one-parameter family y nC1  y n1 C A.y nC1 C .1  2 /y n C y n1 / D ' n 2

(6.30)

with ' n D f .t n ). In these schemes, the transition to a new time level is associated with the inversion of operators E C A, D const . To design additive schemes, we approximate this operator using operators with a simpler structure of type E C A˛ , ˛ D 1, 2, : : : , p. Such a technique is actually implemented in factorized additive schemes. In regularized additive schemes, the basic idea is that we use conditionally stable explicit schemes. To go to unconditionally stable schemes, we modify a problem operator or its individual terms. The explicit scheme belongs to the class (6.30) if  D 0.

177

Section 6.3 Schemes of higher-order accuracy

Unfortunately, such a scheme cannot be taken as the generating one since it is unconditionally unstable and no manipulation via small perturbations of A resolves the situation.

6.3.2 Explicit three-level scheme Among multilevel difference schemes we can select the explicit Adams methods. To start with the generating scheme, we take the explicit three-level scheme 3y n  y n1 y nC1  y n CA D ' n,  2

n D 1, 2, : : :

(6.31)

where ' n D f .t nC1=2 /. First of all, we formulate a condition for the stability of the scheme. The scheme (6.31) may be written in the canonical form for three-level difference schemes: y BQ

nC1

 y n1 Q n D 'n. Q nC1  2y n C y n1 / C Ay C R.y 2

(6.32)

Taking into account 1 1 y nC1  y n D .y nC1  y n1 / C .y nC1  2y n C y n1 /, 2 2 1 1 3y n  y n1 D .y nC1  y n1 /  .y nC1  2y n C y n1 / C 2y n , 2 2 the scheme (6.31) obtains the representation (6.32), where 1     E  A , AQ D A. BQ D E C A, RQ D 2 2 2 The necessary and sufficient condition for stability of the three-level scheme (6.31) with self-adjoint operators (see Theorem 2.10) has the form BQ  0,

1 Q RQ  A, 4

AQ > 0.

(6.33)

By (6.33), the explicit scheme (6.31) is stable under the condition 1 1 1 E  A  0. RQ  AQ D 4 2 2 This gives the constraint on a time step: 

1 . kAk

Consequently, in comparison with the two-level explicit scheme, this reduces the permissible time step by a factor of two.

178

Chapter 6 Regularized additive schemes

6.3.3 Regularized schemes To construct unconditionally stable schemes, we can again focus on the multiplicative regularization of the problem operator A. Remaining in the class of schemes with the second-order approximation, instead of (6.31), we use the regularized scheme 3y n  y n1 y nC1  y n C .E C A2 /1 A D ' n,  2 n D 1, 2, : : : ,

(6.34)

where the regularization parameter  D   2 . The condition for stability of (6.33) holds if E   .E C A2 /1 A D .E C A2 /1 .E C A2  A/    2 2 2 1 A  0. D .E C A / .E  A/ C   2 4 Thus, the regularized scheme (6.34) is unconditionally stable if   0.25 2 (  0.25). This scheme corresponds to the scheme y nC1  y n 3y n  y n1 CA D ' n , n D 1, 2, : : :  2 with the additive regularization of the explicit scheme (6.31), which is characterized by a nonstandard choice of a regularizing operator. We can now proceed to the construction of additive schemes for numerically solving the problem (6.27)–(6.29). In the investigation conducted below, we assume that the inversion of the operators E CA2˛ , ˛ D 1, 2, : : : , p is not much more complicated in comparison with the operators E C A˛ , ˛ D 1, 2, : : : , p, i.e., the problem remains simple. For the generating scheme .E C A2 /

3y n  y n1 y nC1  y n X A˛ C D 'n,  2 ˛D1 p

n D 1, 2, : : : ,

by analogy with (6.34) (by means of the standard replacement of the operators A˛ with .E C   2 A2˛ /1 A˛ ), we construct the regularized additive scheme for the problem (6.27)–(6.29): y nC1  y n X 3y n  y n1 .E C   2 A2˛ /1 A˛ C D ' n,  2 ˛D1 p

n D 1, 2, : : : . (6.35)

Theorem 6.3. For   0.25p 2 and any  > 0, the additive second-order scheme .6.35/ is stable.

179

Section 6.3 Schemes of higher-order accuracy

Proof. It is sufficient to verify the fulfillment of the conditions (6.33). The scheme (6.35) may be represented in the form (6.32), where 1     Q RQ D E  AQ , BQ D E C A, 2 2 2 AQ D

p X

.E C   2 A2˛ /1 A˛ .

˛D1

The second condition in (6.33) may be written as p 1 1 X RQ  AQ D .E  p .E C   2 A2˛ /1 A˛ /  0. 4 2p ˛D1

For each individual term, the inequality E  p .E C A2˛ /1 A˛  0, is valid, as was shown above, if  D   2  0.25p 2  2 . Therefore, the regularized additive scheme (6.35) is stable for   0.25p 2 . It is easy to give the corresponding estimates for stability with respect to the initial data and the right-hand side that prove the convergence of the additive scheme (6.35) with the second order.

6.3.4 Additively averaged scheme We have already noted a deep connection between regularized additive schemes of first-order accuracy and additively averaged schemes. It is interesting to give a similar treatment to additive schemes with the second-order approximation in time. Let us introduce y˛nC1 , ˛ D 1, 2, : : : , p such that y nC1 D

p 1 X nC1 y . p ˛D1 ˛

With this in mind, we write the scheme (6.35) in the form  X p  nC1 p X 3y n  y n1 1 n y˛  y n 2 2 1 C .E C   A˛ / A˛ D ' , p 2 p ˛D1 ˛D1

(6.36)

n D 1, 2, : : : .

Now we specify the auxiliary quantities y˛nC1 , ˛ D 1, 2, : : : , p determining them from the equations .E C   2 A2˛ /

y˛nC1  y n 3y n  y n1 C A˛ D .E C   2 A2˛ /'˛n , p 2

n D 1, 2, : : : ,

˛ D 1, 2, : : : , p,

(6.37)

180

Chapter 6 Regularized additive schemes

where

1 n ' . p We have demonstrated the equivalence of the additively averaged scheme (6.36), (6.37) to the regularized scheme (6.35), which has the second-order approximation by  . For this reason, it is also converges with the second order. '˛n D

6.4 Regularized schemes for equations of second order Now we will demonstrate possibilities of the regularization principle for constructing unconditionally stable additive difference schemes. The standard explicit scheme is used as the generating scheme; its regularization is based on multiplicative perturbation of a problem operator.

6.4.1 Model problem In a finite-dimensional real Hilbert space H , we seek a function u.t / 2 H as the solution of the Cauchy problem for the evolutionary equation of second order: d 2u C Au D f .t /, t > 0, (6.38) dt 2 (6.39) u.0/ D u0 , du (6.40) .0/ D v 0 . dt In the case of positive, self-adjoint, and constant operator A, we apply the splitting AD

p X

A˛ ,

A˛ D A˛  0,

˛ D 1, 2, : : : , p.

(6.41)

˛D1

For the problem (6.38)–(6.40), it is natural to employ the scheme with weights y nC1  2y n C y n1 C A.y nC1 C .1  2 /y n C y n1 / D ' n (6.42) 2 with given y 0 , y 1 , which approximates (6.38) with the second order. The scheme (6.42) may be written in the canonical form for three-level schemes: y BQ where

nC1

 y n1 Q n D ' n, Q nC1  2y n C y n1 / C Ay C R.y 2

1 RQ D 2 E C A, AQ D A.  The condition 1 Q AQ > 0 BQ  0, RQ  A, 4 ensures the stability of the scheme.

(6.43)

BQ D 0,

(6.44)

Section 6.4 Regularized schemes for equations of second order

181

The scheme with weights satisfies the restriction (6.44) if 1 1 .   4 kAk 2 For the explicit scheme, this leads to the restriction on a time step: 1 .  2kAk1=2 For the splitting (6.41), we modify the explicit scheme by means of decreasing kAk in such a way that this condition is always satisfied.

6.4.2 Regularized scheme For the explicit scheme y nC1  2y n C y n1 C Ay n D ' n , 2 the simplest multiplicative regularization yields the scheme y nC1  2y n C y n1 C .E C A/1 Ay n D ' n . (6.45) 2 To maintain the second-order approximation, we choose the regularization parameter  D   2 . In a similar form, we can write the scheme with weights (6.42), which differs from the scheme (6.45) only in the right-hand side. The stability condition (6.44) of the scheme (6.45) leads to the inequality  1 1 1 R  A D 2 E   2 .E C A/1 A  0. 4  4 We have   1 1 E   2 .E C A/1 A D E C A/1 .E C A   2 A 4 4 and therefore, under the constraint   0.25 2 (  0.25), the stability condition is valid. A multicomponent analog of (6.45) leads to the regularized additive scheme. The explicit scheme is taken as the generating scheme, i.e., y nC1  2y n C y n1 X C A˛ y n D ' n . 2 ˛D1 p

The multiplicative perturbation of the individual operator terms gives the desired additive scheme y nC1  2y n C y n1 X C .E C A˛ /1 A˛ y n D ' n . 2 ˛D1 p

(6.46)

182

Chapter 6 Regularized additive schemes

Theorem 6.4. For   0.25p ( D   2 ) and any  > 0, the additive scheme .6.46/ for the problem .6.38/–.6.40/ is unconditionally stable. Proof. Similarly to (6.45), for the scheme (6.46), we have  p  p 1 X 1 X p .E C A˛ /1 A˛ D E  2 E   2 .E C A˛ /1 A˛  0 4 ˛D1 p ˛D1 4 with   0.25p 2 (  0.25p). The regularized additive scheme of multicomponent splitting (6.46) converges under the above restriction on the regularization parameter with the second order by  . The reader can conduct the complete study of this fact without our assistance.

6.4.3 Additively averaged schemes for equations of second order Using the scheme (6.46), we will construct an additively averaged scheme for numerically solving the Cauchy problem (6.38)–(6.40). We have noted a deep connection between regularized additive schemes of first order and additively averaged schemes. It is interesting to give a similar interpretation to additive schemes with the second-order approximation. The auxiliary functions y˛nC1 , ˛ D 1, 2, : : : , p are determined from the equations y˛nC1  2y n C y n1 1 C .E C A˛ /1 A˛ y n D ' n , 2 p p ˛ D 1, 2, : : : , p,

(6.47)

n D 1, 2, : : : .

The following expression is treated as the solution at a new time level: y

nC1

p 1 X nC1 D y . p ˛D1 ˛

(6.48)

The additively averaged scheme (6.47), (6.48) is equivalent to the regularized additive scheme (6.46). For  D   2 , we can reduce this scheme by means of a small perturbation of the right-hand side to the scheme 1 y˛nC1  2y n C y n1 C A˛ .y˛nC1 C .1  2 /y n C y n1 / D ' n , 2 p p ˛ D 1, 2, : : : , p,

n D 1, 2, : : : ,

which has the usual form of a scheme with weights and converges with the second order by  .

Section 6.5 Regularized schemes with general regularizers

183

6.5 Regularized schemes with general regularizers Here (see [202]) regularized additive operator-difference schemes for evolutionary problems are constructed under more general conditions – without the assumption of commutativity of a regularizer and a problem operator. We construct regularized additive schemes with double multiplicative perturbation of additive terms of the problem operator. Special attention is paid to factorized schemes of multicomponent splitting. Possibilities of generalizing the proposed regularized additive schemes of full approximation to problems with non-self-adjoint operators and second-order equations are discussed.

6.5.1 General regularizers A major problem in the construction of stable difference schemes based on the regularization principle is the choice of a regularizing operator R. We have already discussed several constructs related to some perturbation of the scheme operators. To specify a regularizer, we actually used only two possibilities: (6.9) and (6.10). A somewhat more general version assumes that R D R.A/. In fact, the restriction is due to the commutativity of the regularizer R and the operator A, i.e., RA D AR. Consider the construction of regularized operator-difference schemes in the case where operators R and A do not commute. Let us consider the problem du C Au D 0, t > 0 (6.49) dt with the initial condition u.0/ D u0 . (6.50) Assume that the linear operator A acting from H onto H (A : H ! H ) is positive, self-adjoint, and time-independent .A D A > 0, dA=dt D Ad=dt /. Suppose also that RA ¤ AR. (6.51) In constructing regularized schemes, we will use the explicit scheme y nC1  y n C Ay n D 0, 

n D 0, 1, : : : ,

y 0 D u0

(6.52) (6.53)

as the generating scheme. A regularized scheme corresponds to the selection B D E C R. in the scheme B instead of B D E.

y nC1  y n C Ay n D 0, 

(6.54) n D 0, 1, : : :

(6.55)

184

Chapter 6 Regularized additive schemes

To choose a regularizer, it is natural to seek the most simple computational implementation. For example, in [126, 131] for constructing schemes for problems with variable coefficients, a regularizing operator corresponds to the problem with constant coefficients. In this case, the discrete problem at a new time level is greatly simplified. Concerning to the problem (6.49),(6.50), we restrict ourselves, for simplicity, to an important special class of regularizers R D R  A.

(6.56)

Direct computations show that the regularized operator-difference scheme (6.54)– (6.56) is unconditionally stable with   =2. In the construction of regularized additive schemes, to prove the stability of the schemes of type (6.54)–(6.56), we need to establish the stability for each operator term A˛ , ˛ D 1, 2, : : : , p in the decomposition AD

p X

A˛ ,

A˛ D A˛  0,

˛ D 1, 2, : : : , p.

(6.57)

˛D1

To obtain an overall stability estimate from stability conditions for the individual subproblems, we should show stability in a single norm. For the problem (6.49), (6.50) considered here, such a norm corresponds to H . For sufficiently small  , the explicit scheme (6.52), (6.53), as well as the solution of the problem (6.49), (6.50), is stable in any norm of the space HD of grid functions generated by the operator D D D  > 0 if DA D AD. For the regularized scheme (6.54)–(6.56)), stability takes place in the HD with D D A,

 D D E C R  A 2

and some of their combinations. In particular, stability holds for D D E C R. It is essential for us that for the general regularizers (6.51), it is not possible to prove stability in H . For this reason, we cannot focus in the construction of additive operatordifference schemes on the scheme of type (6.54), (6.55) with the general regularizers (6.51). To design regularized operator-difference schemes with regularizers (6.51), (6.56) that are stable in H , we use the regularization A 7! .E C R /1 A.E C R/1 . For the generating scheme (6.52), (6.53) under examination, we set y nC1  y n Q n D 0, C Ay  where

n D 0, 1, : : :

AQ D .E C R/1 A.E C R/1 .

(6.58) (6.59)

185

Section 6.5 Regularized schemes with general regularizers

To obtain the necessary and sufficient stability condition, we employ the inequality  (6.60) B  A. 2 Under the assumption (6.56), it is equivalent to the inequality  .E C R/.E C R/  R  0, 2 which holds for   =8.

6.5.2 Additive schemes with a general-form regularizer Let us construct additive schemes based on the regularization principle for difference schemes. In designing unconditionally stable additive schemes for the problem (6.49), (6.50), (6.57), we use the simplest explicit scheme y nC1  y n X A˛ y n D 0, C  ˛D1 p

n D 0, 1, : : :

(6.61)

as the generating scheme. In the case of the multiplicative regularization with R˛ D A˛ , we will perturb each operator term in (6.57): y nC1  y n X .E C ˛ A˛ /1 A˛ y n D 0, C  ˛D1 p

n D 0, 1, : : : .

(6.62)

For ˛  p=2, ˛ D 1, 2, : : : , p and arbitrary  > 0, the following a priori estimate holds for (6.53), (6.61): (6.63) ky nC1 k  ku0 k. The regularized scheme (6.62) under consideration can be implemented in the form of an additively averaged scheme of full approximation: yQ nC˛=p  y n C A˛ yQ nC˛=p D 0,  p 1 X nC˛=p nC1 D yQ . y p ˛D1

˛ D 1, 2, : : : , p,

(6.64) (6.65)

Note that the construction of such an additive multicomponent splitting scheme is performed without using the concept of full approximation. In addition, the additive scheme (6.64), (6.65) admits asynchronous computations; it is very important for parallel computations. Regularizing the explicit additive scheme (6.61) by means of the technique (6.58), (6.59), we obtain y nC1  y n X Q n C A˛ y D 0,  ˛D1 p

n D 0, 1, : : :

(6.66)

186

Chapter 6 Regularized additive schemes

AQ˛ D .E C ˛ R˛ /1 A˛ .E C ˛ R˛ /1 ,

˛ D 1, 2, : : : , p

(6.67)

with the regularizers R˛ D R˛  A˛

˛ D 1, 2, : : : , p.

(6.68)

The stability condition (6.60) for the scheme (6.66)–(6.68) may be rewritten in the form p p X X 1  Q E A˛ . p 2 ˛D1 ˛D1 For example, these conditions are fulfilled if  1 (6.69) E  AQ˛ . p 2 To ensure the inequality (6.69), we put ˛  p=8, ˛ D 1, 2, : : : , p. Under these restrictions on ˛ , ˛ D 1, 2, : : : , p, the regularized additive scheme (6.66)–(6.68) is stable in H , and its solution satisfies the a priori stability estimate with respect to the initial data (6.63). Considering the problem for the error, we can prove the convergence of the approximate solution to the exact one with a rate of O. /. Similar conclusions can be derived for the additive multicomponent splitting scheme. In this case, we have y nC˛=p  y nC.˛1/=p (6.70) C AQ˛ y nC.˛1/=p D 0, ˛ D 1, 2, : : : , p  if (6.67) is used. The stability condition for the operator-difference scheme (6.67), (6.70) has the form ˛  1=8, ˛ D 1, 2, : : : , p.

6.5.3 Factorized additive schemes To numerically solve problems for evolutionary equations, factorized operator-difference schemes may be very useful. Such schemes are characterized by multiplicative perturbation of the operator at the time derivative. The most interesting examples of such schemes were constructed using the additive splitting of the problem operator into two operator terms (p D 2 in the representation (6.57)). If regularizers from class R D A are used, then the two-component factorized scheme has the form y nC1  y n (6.71) C .A1 C A2 /y n D 0. .E C A1 /.E C A2 /  In the case  D =2, the scheme (6.71) corresponds to the classical Peaceman– Rachford ADI method; in the case  D  , it corresponds to the Douglas–Rachford algorithm. The unconditional stability of the factorized scheme (6.71) is proved for   =2, and the stability estimate is     k E C A2 /y nC1 k  k E C A2 /y n k. i.e., in HD , where D D .E C A2 /2 .

187

Section 6.5 Regularized schemes with general regularizers

For general regularizers from the class (6.56), a factorized scheme of multicomponent splitting is written in the canonical form (6.55) in the case of the additive representation (6.57) with the operator BD

p Y 

E C RpC1˛

p Y   E C Rp .

˛D1

(6.72)

˛D1

Under the adopted assumptions, R˛ D R˛  A˛ ; therefore, B D B  > 0, and we can use the general stability conditions (6.60). Compared with conventional factorized schemes of type (6.71), the computational cost at every time step doubles. A simple modification of the scheme (6.73) based on the operator BD

p1 Y





E C RpC1˛ .E C 2R1 /

˛D1

p Y 

E C Rp



˛D2

slightly decreases the computational costs.

6.5.4 Generalizations Let us discuss some important generalizations of the above results and directions of future studies. First, this concerns problems with non-self-adjoint operators. For A˛ ¤ A˛  0, ˛ D 1, 2, : : : , p, regularized additive schemes can be easily constructed using the regularizes R˛ D A˛ , ˛ D 1, 2, : : : , p. In this case, additive schemes (6.62) and (6.64), (6.65) can be employed. The construction of additive schemes with general regularizers R˛  A˛ , ˛ D 1, 2, : : : , p requires special consideration. This remark concerns both the additive schemes of type (6.66), (6.67) and the factorized schemes (6.55), (6.72). An example of generalization of the Cauchy problem (6.49), (6.50) is the problem where equation (6.49) is replaced with the equation D

du C Au D 0, dt

t >0

(6.73)

with the operator D D D  > 0 such that dD=dt D Dd=dt . The main feature of the problem (6.50), (6.73) is stability with respect to the initial data and the right-hand side in HD . Additive schemes for the problem (6.50), (6.57), (6.73) can be developed based on the transition to the new unknown function v.t / D D 1=2 y.t / that satisfies the equation dv Q D 0, t > 0, C Av dt where AQ D D 1=2 AD 1=2 .

188

Chapter 6 Regularized additive schemes

An analog of the regularized scheme (6.62) is the scheme y nC1  y n X D.D C ˛ A˛ /1 A˛ y n D 0, C D  ˛D1 p

n D 0, 1, : : : .

Similarly, analogs of the other additive schemes considered above can be designed. The construction of additive operator-difference second-order schemes deserves special attention. To this end, we can use the three-level explicit scheme 3y n  y n1 y nC1  y n X A˛ C D 0,  2 ˛D1 p

n D 1, 2, : : : .

(6.74)

(the Adams method) as the generating scheme instead of the two-level explicit scheme (6.61). For the regularizers R˛ D A˛ , ˛ D 1, 2, : : : , p, the second-order regularized additive scheme may be written as 3y n  y n1 y nC1  y n X .E C ˛  2 A2˛ /1 A˛ C D 0,  2 ˛D1 p

n D 1, 2, : : : .

Its stability for ˛  p 2 =4, ˛ D 1, 2, : : : , p is verified straightforwardly. Other variants of regularized additive schemes are designed in a similar fashion. We note also that the Samarskii regularization principle for operator-difference schemes has great potential in constructing additive schemes for the numerical solution of second-order evolutionary equations. We will seek an approximate solution of the Cauchy problem d 2u C Au D 0, dt 2

t > 0,

(6.75)

u.0/ D u0 ,

(6.76)

du .0/ D v 0 . dt The explicit scheme for the problem (6.57), (6.75)–(6.77): y nC1  2y n C y n1 X C A˛ y n D 0, 2 ˛D1

(6.77)

p

n D 1, 2, : : :

has the second-order approximation with respect to  and is stable for  0< . 4kAk The regularization of type (6.62) yields the regularized scheme y nC1  2y n C y n1 X C .E C ˛  2 A˛ /1 A˛ y n D 0, 2 ˛D1 p

n D 1, 2, : : : ,

Section 6.5 Regularized schemes with general regularizers

189

which is unconditionally stable for ˛  p=4, ˛ D 1, 2, : : : , p. For more general regularizers R˛  A˛ , ˛ D 1, 2, : : : , p, we can apply the additive scheme p p X y nC1  2y n C y n1 X 2 1 C .E C   R / A .E C ˛  2 R˛ /1 y n D 0, ˛ ˛ ˛ 2 ˛D1 ˛D1

n D 1, 2, : : : . Additively averaged operator-difference schemes and factorized schemes are constructed in a similar manner.

Chapter 7

Additive schemes based on approximations of a transition operator In the stability theory for operator-difference schemes developed by Samarskii, stability is studied using the canonical form for operator-difference schemes. An alternative possibility, which is traditionally in common use to analyze methods for solving the Cauchy problem for systems of ODEs, is associated with estimating a norm of an operator of the transition from the current time level to the next one. Here we discuss stability of operator-difference schemes for a model operator-differential equation of first order. Main attention is paid to the construction of additive schemes based on approximations of a transition operator. In particular, the classical factorized schemes, schemes of componentwise splitting, and regularized operator-difference schemes are associated with the use of a multiplicative representation of a transition operator. Additively averaged operator-difference schemes employ an additive representation of a transition operator. We investigate possibilities of constructing splitting schemes of second and higher order with respect to time. There are designed as inhomogeneous additive operator-difference schemes, where each individual operator of splitting uses its own distinct transition operator.

7.1 Operator-difference schemes Problems of constructing approximations in time for an evolutionary equation of the first order are discussed. Discretizations based on Padé approximations are studied separately for problems with self-adjoint and skew-symmetric operators. The material generalizes the papers [185, 201, 203, 204].

7.1.1 Operator-differential problem Let H be a finite-dimensional real Hilbert space, where the scalar product and norm are ., / and k  k, respectively. We seek the solution u.t / (0  t  T , T > 0) of the evolutionary first-order equation du C Au D f .t /, 0 < t  T , dt u.0/ D u0 .

(7.1) (7.2)

Here the right-hand side f .t / 2 H in equation (7.1) is given and A is a non-negative, time-independent, and linear operator from H onto H .

191

Section 7.1 Operator-difference schemes

It is easy to obtain a stability estimate for the problem (7.1), (7.2). Taking into account the skew-symmetry of the operator A, we have the equality kuk

d kuk D .f , u/. dt

In view of .f , u/  kukkf k, we get an elementary stability estimate for the problem (7.1), (7.2) with respect to the initial data and the right-hand side: Z t kf . /kd . (7.3) ku.t /k  ku0 k C 0

We would like to inherit this property of the differential problem in discrete analogs of the problem (7.1), (7.2). For the homogeneous problem (f D 0), the norm of the solution is independent of time: (7.4) ku.t /k  ku0 k. In this case, we can write the solution of the problem (7.1), (7.2) in the form u.t / D exp.tA/u0 . The operator exponential may be represented as 1 1 exp.tA/ D E  tA C t 2 A2 C    .t /k Ak C    . 2 kŠ

(7.5)

(7.6)

The stability estimate (7.4) follows from the fact that k exp.tA/k  1 under our assumption that the problem operator is non-negative (A  0). For the inhomogeneous problem (7.1), (7.2), instead of (7.5), we have Z t 0 exp..t   /A/f . /d , (7.7) u.t / D exp.tA/u C 0

and the estimate for stability has the form (7.3). The construction of difference schemes for numerically solving the problem (7.1), (7.2) is based on a particular choice of approximations for the operator exp.tA/ and a quadrature formula for the last term in (7.7) appearing from the right-hand side of equation (7.1).

7.1.2 Difference approximations in time Approximations in time are of great interest in the numerical solution of time-dependent problems of mathematical physics. In particular, special attention is given to computational algorithms of higher accuracy (see, e.g., [55, 63]). Along with increasing the accuracy of approximation in space, the problems of improving the accuracy

192

Chapter 7 Schemes based on approximations of a transition operator

with respect to time are also considered focusing primarily on numerical methods for ODEs [6, 88]. In view of specific features of time-dependent problems for PDEs, first of all, we are interested in methods for the numerical solution of the Cauchy problem for stiff systems of ODEs [18, 58, 109]. To improve accuracy of an approximate solution, various approaches are used for time-dependent problems. Fore two-level schemes involving unknowns at two successive time levels, polynomial approximations of scheme operators for the equation solution are employed in one form or another. The well-known example of such schemes is the class of Runge–Kutta methods [18, 63]; they are the most-used ones in modern computational practice. The main feature of multilevel schemes (multistep methods) appears in the approximation of the time derivatives with higher accuracy using a multipoint stencil. Multistep methods based on backward differentiation formulas [42] provide a typical example. In constructing discrete analogs, we try to inherit the most important properties of the differential problem of mathematical physics being studied [131]. In particular, we highlight the properties of the fulfillment of the relevant conservation laws at the discrete level; such a scheme is called a conservative scheme. The second example is a monotone difference scheme; the difference solution of a monotone scheme satisfies the maximum principle. For time-dependent problems, emphasis is on the main attribute of well-posedness of discrete problems, namely, on the stability of the numerical solution with respect to small perturbations of the initial data, boundary conditions, right-hand sides, and coefficients of equations. Various classes of stable difference schemes can be constructed for the numerical solution of PDEs. The topical problem is choosing from among stable difference schemes the scheme that is optimal with respect to some additional criteria. In the theory of difference schemes, we emphasize a class of asymptotically stable schemes that guarantee the correct behavior of the approximate solutions for large time values [134, 142]. In the theory of numerical methods for solving systems of ODEs [18, 42], the concept of L-stable methods is introduced in which the asymptotic behavior of approximate solutions at large time values is also monitored in some distinct sense. In the work [203], the questions of the choice of approximation in time are discussed using an elementary boundary value problem for a one-dimensional parabolic equation as an example. Demands on unconditionally stable difference schemes are formulated; they are associated with the inheritance of the basic properties of differential problems. The concept of SM-stability is introduced for difference schemes. Difference schemes derived on the basis of various Padé approximants are selected as the generating schemes. To solve numerically the homogeneous (f .t / D 0) problem (7.1), (7.2), we employ two-level difference schemes. Define the uniform grid in time with a step  as !  D ! [ ¹T º D ¹t n D n ,

n D 0, 1, : : : , N0 ,

N0 D T º

193

Section 7.1 Operator-difference schemes

and set y n D y.t n /, where t n D n . We want to pass from the time level t n to the next time level t nC1 . The exact solution has the form u.x, t nC1 / D exp.A/u.x, t n /.

(7.8)

The two-level scheme for the problem considered here may be written in the following canonical form (see [131]), i.e., ey B

 yn e n C Ay D 0, 

nC1

n D 0, 1, : : : ,

(7.9)

e are certain operators. The homogeneous scheme (7.9) may be written where e A and B in the form y nC1 D Sy n , n D 0, 1, : : : , (7.10) where

S D E   BQ 1 AQ

(7.11)

is the transition operator from one time level to the next one. In the general case, the operator S may depend on n. We restrict our consideration to the difference approximations with respect to time for the homogeneous problem (7.1), (7.2) that lead to the transition operator S D s.A/,

(7.12)

where s.z/ is a certain function. In the stability theory for finite difference schemes developed by Samarskii [131,134,136], necessary and sufficient stability conditions in various Hilbert spaces of grid functions were obtained for self-adjoint operators A and B; generalizations to the case of non-self-adjoint operators using the corresponding operator inequalities are also known. Under the restriction (7.12), i.e., for B D B.A/, stability conditions in Hilbert spaces are easily verified using the properties of the function s.z/ only. The following main result holds true. Let A  ıE. Then we have ks.A/k  max js.z/j, Re zı

where it is not assumed that A is self-adjoint. Such a technique is standard in the theory of the numerical methods for the Cauchy problem for systems of ODEs; here s.z/ is called a stability function [18, 58]. For the standard scheme with weights y nC1  y n C A.y nC1 C .1   /y n / D 0,  the transition operator may be represented as

n D 0, 1, : : : ,

S D .E C  A/1 .E C .  1/A/.

(7.13)

(7.14)

194

Chapter 7 Schemes based on approximations of a transition operator

The scheme (7.10), (7.14) is stable if kS k  1.

(7.15)

The inequality for the norm (7.15) is equivalent to the operator inequality S S   E. By (7.14), we have .E C  A/1 .E C .  1/A/.E C .  1/A /.E C  A /1  E. Multiplying this inequality from the left by .E CA/ and from the right by .E CA /, we obtain 2A C .2  1/ 2 AA  0. If A  0, then stability is ensured by a choice of   0.5.

7.1.3 SM-stable schemes for problems with a self-adjoint operator We connect a choice of approximations in time with two various classes of problems: with a self-adjoint operator and with a skew-symmetric operator. We start with solving the Cauchy problem for the equation du C Au D 0, dt

0 < t  T,

(7.16)

where A D A  ıE, ı  0. The use of additional information about properties of a problem operator allows to specify requirements for the choice of approximations in time. Taking into account the inequality A  ıE, we get a more accurate stability estimate ku.x, t nC1 /k  exp.ı /ku.x, t n /k. The inheritance of this property of the semi-discrete problem (7.2), (7.16) means that, instead of (7.4), the following estimate must be true: ky nC1 k  ky n k,

ks.A/k  ,

 D exp.ı /.

(7.17)

In this case, we say [134] that this operator-difference scheme is -stable. The properties discussed above concern the bounds of the spectrum of the transition operator. We want also to keep track of some other qualitative behaviors of the approximate solution. In solving unsteady problems, we focus on the long-time asymptotic behavior of the solution. For the model problem under examination, the solution decreases to zero as t ! 1. If the approximate solution retains this property, we say that the approximate solution is asymptotically stable (with respect to time). In the case of

195

Section 7.1 Operator-difference schemes

the Cauchy problem for systems of ODEs, this property of the approximate solution is called L-stability. If (7.18) lim s. / D 0, !1

the stable (-stable) difference scheme (7.9) is said to be asymptotically stable. For the linear problem (7.2), (7.16) with a self-adjoint positive definite operator A, the solution may be written as the superposition of individual harmonics that are associated with their own eigenvalues. In [203], the choice of approximations in time is subjected to the requirement of the appropriate behavior in time of all points of the spectrum. In this case, we introduce the SM-property (Spectral Mimetic) of schemes, i.e., such a scheme is said to be SM-stable. A difference scheme is called SM-stable if it is -stable and asymptotically stable. The additional requirement is the spectral monotonicity – the function s. / is monotonically decreasing. This means that the harmonics with large indexes decay more rapidly than the harmonics with small indexes. Two-level schemes of higher-order approximations for time-dependent linear problems can be conveniently constructed on the basis of Padé approximations of the operator (matrix) exponential function exp.A/. Other approaches are discussed in [60,99]. In the case of nonlinear systems of ODEs, such approximations correspond to various variants of Runge–Kutta methods [18, 58, 63]. The Padé approximation of the function exp.z/ is exp.z/ D Rlm .z/ C O.z lCmC1 /,

Rlm .z/ 

Plm .z/ , Qlm .z/

(7.19)

where Plm .z/ and Qlm .z/ are polynomials of degrees l and m, respectively. These polynomials have [10] the form Plm .z/ D

l X lŠ .l C m  k/Š .z/k , .l C m/Š kŠ.l  k/Š kD0

Qlm .z/ D

m X mŠ .l C m  k/Š k z . .l C m/Š kŠ.m  k/Š kD0

As for the homogeneous equation (7.16), the difference scheme based on Padé approximations corresponds to the two-level scheme y nC1  y n 1 C .Qlm .A/  Plm .A//y n D 0, n D 0, 1, : : : . (7.20)   The scheme (7.20) is written in the canonical form (7.9) with 1 e D Qlm .A/. e (7.21) A D .Qlm .A/  Plm .A//, B  In the simplest case m D 1, we have 1 R01 .z/ D D exp.z/ C O.z 2 /, 1Cz Qlm .A/

196

Chapter 7 Schemes based on approximations of a transition operator

R11 .z/ D

1  12 z D exp.z/ C O.z 3 /. 1 C 12 z

In this case, the approximation R01 .z/ is associated with the fully implicit scheme y nC1  y n C Ay nC1 D 0, 

n D 0, 1, : : :

(7.22)

for the numerical solution of the problem (7.2), (7.16). Similarly, the approximation R11 .z/ correspond to the symmetric (Crank–Nicolson) scheme y nC1 C y n y nC1  y n CA D 0,  2

n D 0, 1, : : : .

(7.23)

Let us turn our attention to the SM-properties of these classical schemes. Figure 7.1 shows the plots of the functions R11 .z/, R11 .z/, and exp.z/. It is easy to see that the scheme (7.22) is SM-stable while the scheme (7.23) is not. For the symmetric scheme, the asymptotic stability condition does not hold because, in this case, we have lim s. / D 1, s. / D R11 . /. !1

In addition, there are problems (see, e.g., [134, 142]) with the unconditional validity of the -stability property.

exp(−z) R01 (z) R11 (z)

1

f (z)

0.5

0

−0.5

−1 −2

0

2

4

6

8

10 z

Figure 7.1. The Padé approximation with m D 1.

12

14

16

18

20

22

197

Section 7.1 Operator-difference schemes

In computational practice, the implicit scheme (7.22) is often preferable to the scheme (7.23) even though the symmetric scheme has a higher (the second) approximation order. We can attribute the advantages of the fully implicit scheme to its SMstability, i.e., to the more correct inheritance of the spectral characteristics of the solution. We seek SM-stable schemes based on Padé approximations for m > 1. In the case m D 2, we have R02 .z/ D

1 D e z C O.z 3 /, 1 C z C 12 z 2

(7.24)

R12 .z/ D

1  13 z D e z C O.z 4 /, 1 C 23 z C 16 z 2

(7.25)

R22 .z/ D

1 2 1  12 z C 12 z z C O.z 5 /. 1 1 2 De 1 C 2 z C 12 z

(7.26)

Figure 7.2 demonstrates the plots of the functions R02 .z/, R12 .z/, R22 .z/, and exp.z/. For the approximation R22 .z/, the asymptotic stability and spectral monotonicity conditions do not hold. The schemes based on the approximation R12 .z/ are asymptotically stable but not spectrally monotone. In this case, the violation of the latter property results in considerable qualitative inconsistencies in the approximate solution; namely, certain harmonics grow with time while they should decay.

exp(−z) R02 (z) R12 (z) R22 (z)

1 0.8

f (z)

0.6 0.4 0.2 0 −0.2 −2

0

2

4

6

8

Figure 7.2. The Padé approximation with m D 2.

10 z

12

14

16

18

20

22

198

Chapter 7 Schemes based on approximations of a transition operator

Such behavior is also typical for other schemes based on Padé approximations. Only the schemes with l D 0, i.e., 1 R0m .z/  , P0m .z/ D 1 Q0m .z/ are SM-stable. In this case, the difference scheme for the problem (7.2), (7.16) is ynC1  yn 1 (7.27) C .Q0m .ƒ/  E/yn D 0, n D 0, 1, : : : , Q0m .ƒ/   where the function m X 1 k (7.28) z Q0m .z/ D kŠ kD0

is a truncated Taylor series for exp.z/. Figure 7.3 presents the corresponding stability functions for m D 1, 2, 3. exp(−z) R01 (z) R02 (z) R03 (z)

1 0.8

f (z)

0.6 0.4 0.2 0 −2

0

2

4

6

8

10 z

12

14

16

18

20

22

Figure 7.3. SM-stable Padé approximations.

Let us introduce the concept of conditionally SM-stable schemes. If, for 0 <   0 , the scheme (7.9) for the problem (7.2), (7.16) is %-stable and asymptotically stable, then it is said to be conditionally SM-stable. The symmetric scheme (7.23) (see Figure 7.1) is SM-stable for 0 corresponding to z0 such that R11 .z0 / D 0. Similarly, the conditional SM-stability conditions for the schemes based on Padé approximations with m D 2 can be found. The scheme with R22 .z/ (see Figure 7.2) is SM-stable if z0 minimizes R22 .z/. The conditional stability conditions for the schemes are only slightly less strong than those for the explicit scheme; therefore, they can hardly be useful in practice.

199

Section 7.1 Operator-difference schemes

7.1.4 Factorized SM-stable two-level schemes The main drawback of the above SM-stable schemes based on Padé approximations is related to its computational implementation. The transition to a new time level requires that a matrix polynomial must be inverted. The computational costs of this operation are significantly higher in comparison with the simplest schemes with weights. In terms of computational implementation of the schemes of higher order for the numerical solution of the initial-boundary value problems for parabolic equations, the most convenient are factorized schemes. In this case, diagonally implicit Runge–Kutta (DIRK methods) [18, 63] appear for systems of ODEs. The work [201] concerns the construction of just such factorized SM-stable schemes. It would be natural to approximate exp.z/ by a ratio of polynomials of type (7.19) choosing as the denominator the polynomial Qlm .z/ D .1 C lm z/m ,

(7.29)

where lm > 0 is a constant. The approximation given by (7.19), (7.29) was proposed in [101]; it is called the RD-Padé approximation (where RD means Restricted Denominator). If m D 1, then the RD-Padé approximation is identical to the conventional Padé approximation and is related to the use of fully implicit (l D 0) and symmetric (l D 1) schemes. In view of the asymptotic stability condition (7.18), we restrict ourselves to the case l D m  1. Define Gm .z/ D Rlm .z/. We illustrate the construction of the RD-Padé approximations by the simplest example, where m D 2. In this case, we have G2 .z/ D

1 C z , .1 C  z/2

where  D 12 . The coefficients  and  are found from the best approximation condition for exp.z/. We have   1 2 2 3 .1 C  z/ D .1 C z/ 1 C z C z C O.z / , 2 which yields the system of equations 2 D  C 1, 1 2 D  C . 2 Its solutions are two pairs  and  , i.e., p p 2 1 D 1  , 1 D 1  2, 2 p p 2 2 D 1 C , 2 D 1 C 2. 2

(7.30) (7.31)

200

Chapter 7 Schemes based on approximations of a transition operator

The RD-Padé approximation

p 1 C .1  2/z D e z C O.z 3 /, p G2 .z/ D .1 C .1  2=2/z/2 which corresponds to the selection (7.30), does not yield an unconditionally SM-stable scheme p because the spectral monotonicity condition is not fulfilled – for z D z0 D 1=. 2  1/, we have G2 .z0 / D 0. For the pair (7.31), we obtain p 1 C .1 C 2/z D e z C O.z 3 /. (7.32) p G2 .z/ D .1 C .1 C 2=2/z/2 This approximation leads to an unconditionally SM-stable scheme. exp(−z) σ = σ1 σ = σ2 σ = σ3

1 0.8

f (z)

0.6 0.4 0.2 0 −0.2 −2

0

2

4

6

8

10 z

12

14

16

18

20

Figure 7.4. Factorized approximations with m D 3.

By analogy with (7.32), we have 1 C 1 z C 2 z 2 . .1 C  z/3 The coefficient  is determined from the cubic equation 3 1  3  3 2 C   D 0, 2 6 whose roots are G3 .z/ D

1 D 0.158983899988677,

2 D 0.435866521508459,

3 D 2.405149578502866.

22

201

Section 7.1 Operator-difference schemes

σ σ σ σ

1

= σ1 = σ2 = σ3 = σ4

f (z)

0.5

0

−0.5

−1 −2

0

2

4

6

8

10 z

12

14

16

18

20

22

Figure 7.5. Factorized approximations with m D 4.

The approximating functions corresponding to these three root sets are depicted in Figure 7.4. SM-stable schemes are obtained for  D 3 only, where G3 .z/ D

1 C 6.215448735508597z C 10.638784749408941z 2 . .1 C 2.405149578502866z/3

In the case where m D 4, a similar analysis results in the quartic equation 2 1 D 0,  4  4 3 C 3 2   C 3 24 which has four real roots 1 D 0.106438792142663,

2 D 0.220428410259214,

3 D 0.572816062482134,

4 D 3.100316735115990.

The functions G4 .z/ corresponding to these roots are plotted in Figure 7.5. For  D 1 , 2 , 3 , the function G4 .z/) is not monotone. An unconditionally SMstable scheme is obtained for G4 .z/ with  D 4 , i.e., G4 .z/ D .1 C 3.100316735115990z/4 .1 C 11.401266940463961z C 45.770516207777675z 2 C 67.562713280972929z 3 /, which approximates the function exp.z/ with O.z 5 /. In Figure 7.6, we show the stability functions for the unconditionally SM-stable schemes based on the RD-Padé approximations. Increasing the approximation order

202

Chapter 7 Schemes based on approximations of a transition operator

exp(−z) G3 (z) G3 (z) G4 (z)

1 0.8

f (z)

0.6 0.4 0.2 0 −2

0

2

4

6

8

10 z

12

14

16

18

20

22

Figure 7.6. SM-stable factorized approximations.

only slightly improves accuracy for z > 1. These schemes are significantly inferior to SM-stable schemes based on Padé approximations (compare Figure 7.6 with Figure 7.3). The qualitative distinctions between these types of schemes can be formally characterized by the value of the leading coefficient of the truncation error. We set R0m .z/ D exp.z/ C CR z mC1 C O.z mC2 /, Gm .z/ D exp.z/ C CG z mC1 C O.z mC2 /. The values of the constants CR and CG are presented in Table 7.1. Table 7.1. The leading term of the truncation error. m

CR

2 3 4

0.166666666666667  0.041666666666667 0.008333333333333

CG 1.373773447853214 6.396967163955937 41.876495464301705

These data demonstrate that RD-Padé approximations suffer from a catastrophic loss of accuracy (revealed by the fact that jCG j is much greater than jCR j). This largely reduces their potential advantages over conventional Padé approximations in the sense of numerical implementation, although, formally, the approximation order is preserved.

203

Section 7.1 Operator-difference schemes

7.1.5 Problems with a skew-symmetric operator Now consider the problem (7.2), (7.16), where the operator A is skew-symmetric, i.e., A D A . In this case, the norm of the solution does not vary in time: ku.t /k D ku0 k.

(7.33)

The equality (7.33) reflects the conservatism of the solution (conservation law), i.e., its neutral stability. A difference scheme for the problem (7.2), (7.16), where the operator A D A , is said to be SM-stable [204] if this scheme is neutrally stable. Two-level schemes of higher-order approximation for unsteady linear problems, as in the case of problems with self-adjoint operators, will be constructed using Padé approximations for the operator (matrix) exponential exp.A/ (see (7.19)).

Figure 7.7. The norm of the transition operator.

The neutral stability condition ky nC1 k D ky n k for the two-level scheme (7.10) is satisfied if kS k D 1. In view of (7.12), for A D A , this corresponds to the condition js.z/j D jRlm .z/j D 1,

Re z D 0.

For the fully implicit scheme (7.22), we have jR01 .z/j D p

1 , 1 C y2

z D iy.

(7.34)

204

Chapter 7 Schemes based on approximations of a transition operator

Therefore, the neutral stability condition is not fulfilled, and the fully implicit scheme is not SM-stable in the case of problems with a skew-symmetric operator. In contrast, for the symmetric scheme (7.23), we get jR11 .z/j D 1,

z D iy,

and therefore it is SM-stable for the class of problems under examination. Similar conclusions can be drawn for the schemes based on the Padé approximations with m D 2. Only the scheme based on the approximation R22 is SM-stable. This fact is illustrated in Figure 7.7, where jR11 .z/j is plotted for z D iy. Using the Padé approximations with l < m (see, e.g., Figure 7.7), the amplitude of the harmonics jR01 .z/j < 1 for z D iy, y ¤ 0, and the scheme acquires dissipative properties due to approximation in time. Only for l D m, is the corresponding scheme neutrally stable. The main conclusion of this study is that for the numerical solution of problems with skew-symmetric operators, it is necessary to use approximations in time that are based on the Padé approximant Rmm .z/. Recall that for problems with a self-adjoint operator, the schemes based on the Padé approximation R0m .z/ are preferable.

7.2 Additive schemes with a multiplicative transition operator We discuss splitting schemes where an operator of the transition to a new time level has a multiplicative form. This class of additive schemes includes the usual schemes of componentwise splitting.

7.2.1 Operator-differential problems Additive operator-difference schemes will be constructed using the Cauchy problem du C Au D f .t /, dt

t > 0,

u.0/ D u0 .

(7.35) (7.36)

For simplicity, we restrict ourselves to the case of a constant operator A that satisfies the additive representation AD

p X

A˛ ,

A˛  0,

˛ D 1, 2, : : : , p.

(7.37)

˛D1

For the constant operators A˛ , ˛ D 1, 2, : : : , p, pairwise commutativity is not assumed.

Section 7.2 Additive schemes with a multiplicative transition operator

205

In view of (7.37), the solution of the problem (7.35), (7.36) satisfies the representation     Z t p p X X u.t / D exp  t A˛ u 0 C exp  .t   / A˛ f . /d . (7.38) 0

˛D1

˛D1

To demonstrate designing additive schemes, it is convenient to consider the homogeneous problem (f .t / D 0 in (7.35)). In this case, from (7.38), during the transition to a new time level (from t n to t nC1 ), we have   p X nC1 u.t / D exp   A˛ u.t n /. (7.39) ˛D1

Thus, the construction of various additive schemes is associated with a choice of approximations of the operator exponential for the operator A on the basis of the individual terms A˛ , ˛ D 1, 2, : : : , p. For the two-component splitting (p D 2 in (7.37)), we have exp. .A1 C A2 / D E   .A1 C A2 / 1 C  2 .A21 C A1 A2 C A2 A1 C A22 / C O. 3 /, 2 exp.A2 / exp.A1 / D E   .A1 C A2 / 1 C  2 .A21 C 2A2 A1 C A22 /2 C O. 3 /. 2 For this reason, we get 1 exp. .A1 C A2 // D exp.A2 / exp.A1 / C  2 ŒA1 , A2  C O. 3 /. 2

(7.40)

Here ŒA1 , A2  D A1 A2  A2 A1 is the commutator of the operators A1 and A2 . For the general multicomponent splitting (7.37), similarly to (7.40), we have   p p X Y exp   A˛ D exp.ApC1˛ / C O. 2 /. (7.41) ˛D1

˛D1

In the special case of pairwise commutative operators, we obtain   p p X Y exp   A˛ D exp.ApC1˛ /, ˛D1

A˛ Aˇ D Aˇ A˛ ,

˛D1

˛, ˇ D 1, 2, : : : , p.

206

Chapter 7 Schemes based on approximations of a transition operator

An approximate solution of the problem (7.35)–(7.37) with f .t / D 0, in view of (7.39) and (7.41), can be constructed using v.t nC1 / D

p Y

exp.ApC1˛ /v.t n /.

(7.42)

˛D1

The implementation of (7.42) can be performed via solving the following simpler problems: dv˛ C A˛ v˛ D 0, t n < t  t nC1 , dt ˛ D 1, 2, : : : , p, v1 .t n / D vp .t n /,

v1 .0/ D u0 ,

v˛ .t n / D v˛1 .t nC1 /,

˛ D 2, 3, : : : , p.

The fundamental point is associated with the multiplicative representation of the operator for the approximate solution in (7.42).

7.2.2 Componentwise splitting schemes The difference scheme for solving the problem (7.35)–(7.37) with the right-hand side f .t / D 0 may be written as y nC1 D Sy n ,

n D 0, 1, : : : .

(7.43)

In accordance with (7.39), (7.41), the transition operator is taken in the factorized form SD

p Y

SpC1˛ .

(7.44)

˛D1

The implementation of this componentwise splitting scheme can be carried out on the basis of the sequence y nC˛=p D S˛ y nC.˛1/=p ,

˛ D 1, 2, : : : , p,

n D 0, 1, : : : .

(7.45)

Before we specify the choice of S˛ , ˛ D 1, 2, : : : , p, let us formulate the general result on the convergence of the approximate solution of the scheme (7.43) to the solution of the homogeneous problem (7.35)–(7.37). Theorem 7.1. Let in the scheme .7.43/ kS k  1

(7.46)

exp.A/  S D  mC1 R, R D O.1/.

(7.47)

and let

Section 7.2 Additive schemes with a multiplicative transition operator

207

Then the error for the approximate solution of .7.35/–.7.37/: z n D u.t n /  y n with y 0 D u0 satisfies the estimate kz nC1 k   m

n X

 kRu.t k /k.

(7.48)

kD0

Proof. The error is determined from the operator-difference scheme z nC1 D S z n C u.t nC1 /  S u.t n /. By (7.47), we have u.t nC1 /  S u.t n / D  mC1 Ru.t n /. Taking into account the stability condition (7.46), we obtain the levelwise estimate kz nC1 k  kz n k mC1 kRu.t n /k. Hence, in view of z 0 D 0, it follows that the required estimate (7.48) holds. The a priori estimate (7.48) ensures the convergence of the operator-difference scheme (7.43) with the order m under the conditions of the scheme’s stability (the estimate (7.46)) and the approximation of the transition operator (the estimate (7.47)). Taking into account (7.41), in the additive componentwise splitting scheme (7.43), (7.44), we may expect convergence with the first order. For this, it is enough to require the fulfillment of exp.A˛ /  S˛ D  2 R˛ ,

R˛ D O.1/,

˛ D 1, 2, : : : , p.

(7.49)

The stability condition (7.46) for the additive scheme (7.43), (7.44) holds for kS˛ k  1,

˛ D 1, 2, : : : , p.

(7.50)

Now we can choose the operators S˛ , ˛ D 1, 2, : : : , p. It is natural to start with the standard scheme with weights (7.13). Similarly to (7.14), we set (7.51) S˛ D .E C ˛ A˛ /1 .E C .˛  1/A˛ /. In a particular case, we can use for the individual operators S˛ , ˛ D 1, 2, : : : , p the same weights (˛ D  , ˛ D 1, 2, : : : , p ). The stability condition (7.50) holds under the usual restrictions ˛  0.5, ˛ D 1, 2, : : : , p. The implementation of the additive componentwise splitting scheme (7.43), (7.44), (7.51) for the numerical solution of the problem (7.35)–(7.37) with the homogeneous right-hand side (f .t / D 0) by the scheme (7.45) gives: y nC˛=p  y nC.˛1/=p C A˛ .˛ y nC˛=p C .1  ˛ /y nC.˛1/=p / D 0,  ˛ D 1, 2, : : : , p, n D 0, 1, : : : .

(7.52)

208

Chapter 7 Schemes based on approximations of a transition operator

Thus, we come to the additive operator-difference scheme of full approximation (5.22). We conducted our study of additive schemes based on the approximation of the transition operator using, for simplicity, the homogeneous problem (f .t / D 0) in (7.35)). Inhomogeneous equations are treated in a similar fashion. In this case, instead of (7.39), we apply the representation  u.t

nC1

/ D exp



p X

 Z n A˛ u.t / C

˛D1



t nC1 tn

exp

 .t

nC1

 /

p X

 A˛ f . /d .

˛D1

(7.53) For an approximate calculation of the integral term, we can use some quadrature formula. Taking into account (7.49), it is sufficient to approximate the last term in (7.53) with an accuracy of O. 2 /. Suppose, e.g., Z



t nC1 tn

exp

 .t

nC1

 /

p X

 A˛ f . /d D f .t nC1 / C O. 2 /.

˛D1

Instead of (7.43), we receive y nC1 D Sy n C  ' n ,

n D 0, 1, : : : ,

(7.54)

where ' n D f .t nC1 /. The computational implementation of the additive scheme (7.44), (7.54), similarly to (7.52), gives y nC˛=p  y nC.˛1/=p C A˛ .˛ y nC˛=p C .1  ˛ /y nC.˛1/=p / D '˛n ,  ˛ D 1, 2, : : : , p,

(7.55)

n D 0, 1, : : : .

For the split right-hand sides, we have ² 0, ˛ D 1, 2, : : : , p  1, n '˛ D ' n , ˛ D p.

(7.56)

The additive operator-difference scheme (7.55), (7.56) may be interpreted as a special variant of the scheme of summarized approximation (5.22).

7.3 Splitting schemes with an additive transition operator Here we study splitting schemes with an additive form of the operator of the transition to a new time level. This class of schemes is associated with additively averaged schemes of componentwise splitting as well as with some regularized additive operator-difference schemes.

Section 7.3 Splitting schemes with an additive transition operator

209

7.3.1 Additive approximation of a transition operator Let us consider the Cauchy problem (7.35), (7.36) with an additive representation of the operator of the problem (7.37). We have already investigated splitting schemes that are based on a multiplicative approximation for the transition operator of type (7.41). Special attention should be given to features that provide additive approximations. In this case, we have   p p X 1 X A˛ D exp.pA˛ / C O. 2 /. (7.57) exp   p ˛D1 ˛D1 In numerically solving the homogeneous problem (7.35)–(7.37) (f .t / D 0 in (7.35)), we obtain for the approximate solution based on (7.57) the representation v.t nC1 / D

p 1 X exp.pA˛ /v.t n /. p ˛D1

(7.58)

From (7.58), we can separate the following simpler problems: dv˛ C pA˛ v˛ D 0, dt v˛ .t n / D v.t n /, After that, we put v.t nC1 / D

t n < t  t nC1 , ˛ D 1, 2, : : : , p.

p 1 X v˛ .t nC1 /. p ˛D1

We emphasize an important point of this implementation, which can be used in the construction of algorithms for parallel computing systems. Namely, the individual subproblems for finding v˛ .t nC1 /, ˛ D 1, 2, : : : , p can be solved independently.

7.3.2 Additive schemes To solve numerically the problem (7.35)–(7.37) with the right-hand side f .t / D 0, we will use the scheme (7.39), where the transition operator according to (7.57) is represented in the additive form SD

p 1 X S˛ . p ˛D1

(7.59)

A possible implementation of the scheme (7.39), (7.59) with an additive representation of the transition operator can be performed via the problems y˛nC1 D S˛ y n ,

˛ D 1, 2, : : : , p.

(7.60)

210

Chapter 7 Schemes based on approximations of a transition operator

The approximate solution at a new time level is y

nC1

p 1 X nC1 D y , p ˛D1 ˛

n D 0, 1, : : : .

(7.61)

For schemes with the first order of convergence, we set exp.pA˛ /  S˛ D  2 R˛ ,

R˛ D O.1/,

˛ D 1, 2, : : : , p.

(7.62)

If the inequalities kS˛ k  1,

˛ D 1, 2, : : : , p

(7.63)

hold, then the additive scheme (7.43), (7.59) is stable. Concerning the usual schemes with weights, we can apply S˛ D .E C ˛ pA˛ /1 .E C .˛  1/pA˛ /.

(7.64)

The stability condition (7.63) is valid for ˛  0.5, ˛ D 1, 2, : : : , p. For numerically solving the problem (7.35)–(7.37) with the homogeneous righthand side (f .t / D 0), the additive scheme (7.60), (7.61), (7.64) corresponds to the application of the scheme y˛nC1  y n (7.65) C pA˛ .˛ y˛nC1 C .1  ˛ /y n / D 0, ˛ D 1, 2, : : : , p.  The scheme (7.61), (7.65) is just an additively averaged scheme of componentwise splitting. To take into account the inhomogeneity of the right-hand side of equation (7.35), we can employ various approaches. In this case, instead of (7.43), we have equation (7.54), where, e.g., ' n D f .t nC1 /. The first variant involves the use of y nC1 D

p 1 X nC1 y C  'n, p ˛D1 ˛

n D 0, 1, : : :

(7.66)

instead of (7.61). The homogeneous problems (7.65) are solved for the auxiliary variables. The second approach is based on solving inhomogeneous problems for the auxiliary functions y˛nC1 , ˛ D 1, 2, : : : , p. Instead of (7.65), we use y˛nC1  y n C pA˛ .˛ y˛nC1 C .1  ˛ /y n / D p'˛n ,  For the right-hand sides, we put n

' D

p X ˛D1

'˛n .

˛ D 1, 2, : : : , p.

(7.67)

(7.68)

211

Section 7.4 Further additive schemes

In view of (7.61), (7.67), (7.68), we obtain y nC1 D Sy n C  'Q n ,

n D 0, 1, : : : ,

where 'Q n D

p X

.E C ˛ pA˛ /1 '˛n D ' n C O. /.

˛D1

The convergence of the additive schemes (7.65), (7.66) and (7.61), (7.67), (7.68) with the first order is established in the same way as was done for the schemes with the multiplicative approximation of the transition operator; it is based on Theorem 7.1.

7.3.3 Regularized additive schemes Many regularized additive schemes may be treated as additive schemes that are based on additive approximations for the transition operator. To illustrate this situation, let us consider a typical example. We will study the model problem (7.35)–(7.37). The regularized additive scheme for this problem has (see, e.g., (6.22)) the form y nC1  y n X .E C ˛ pA˛ /1 A˛ y n D ' n , C  ˛D1 p

n D 0, 1, : : : .

(7.69)

The distinction from (6.22) results from a special choice of perturbation parameters (regularizers). The scheme (7.69) may be written in the form of (7.54), where the operator of the transition to a new time level is p 1 X S˛ . S D E   .E C ˛ pA˛ / A˛ D p ˛D1 1

In this case, the operator terms S˛ , ˛ D 1, 2, : : : , p are determined according to (7.64). Thus, we get the above-examined additive scheme (7.54), (7.59), (7.64).

7.4 Further additive schemes We analyze possibilities of constructing additive schemes with second-order accuracy for the approximation of a transition operator. A multiplicative construction of a transition operator is also observed for the classical factorized schemes of two-component splitting.

212

Chapter 7 Schemes based on approximations of a transition operator

7.4.1 Schemes of the second order It is easy to construct additive schemes of second-order accuracy. In this case, we cannot apply the approximation (7.41). Simple calculations show that for the twocomponent splitting, we have     1 1 exp. .A1 C A2 // D exp  A1 exp.A2 / exp  A1 C O. 3 /. 2 2 (7.70) Moreover, such a structure of type (7.70) (see [170]) can also be used for the general multicomponent splitting  exp



p X ˛D1

 A˛

D

p Y ˛D1

 exp

1  A˛ 2

Y p

  1 exp  ApC1˛ C O. 3 /. 2 ˛D1 (7.71)

The implementation of such schemes may be based on the following simpler problems: 1 dv˛ C A˛ v˛ D 0, t n < t  t nC1 , dt 2 ˛ D 1, 2, : : : , p, dv˛ 1 C A2pC1˛ v˛ D 0, t n < t  t nC1 , dt 2 ˛ D p C 1, p C 2, : : : , 2p, v1 .0/ D u0 ,

v1 .t n / D v2p .t n /,

v˛ .t n / D v˛1 .t nC1 /,

˛ D 2, 3, : : : , 2p.

Thus, the calculations are organized as the cycle A1 7! A2 7!    Ap 7! Ap 7! Ap1 7!    A1 , which we have previously noted in considering schemes of summarized approximation. In designing additive schemes, we have to use the approximations exp.A˛ /  S˛ D  3 R˛ ,

R˛ D O.1/,

˛ D 1, 2, : : : , p

(7.72)

for the individual operator factors in (7.44), (7.71). In the simplest class of the schemes with weights, we set  1   1 1 S˛ D E C A˛ E C A˛ . (7.73) 2 2

213

Section 7.4 Further additive schemes

To take into account the inhomogeneity of the right-hand side of equation (7.35), it is possible to apply the trapezoidal quadrature formula. It gives   Z t nC1 p X nC1 exp  .t  / A˛ f . /d tn

˛D1

1 D  .f .t nC1 / C .E  A/f .t n // C O. 3 /. 2 The use of the quadrature formula of rectangles corresponds to Z t nC1 p    X 1  exp  .t nC1   / A˛ f . /d D  E  A f .t nC1=2 / C O. 3 /. 2 tn ˛D1 Therefore, in (7.54), we put 1 ' n D .f .t nC1 / C .E  A/f .t n //, 2   1 (7.74) ' D E  A f .t nC1=2 /. 2 The additive schemes designed above using an approximation of a transition operator lead to additive schemes of summarized approximation. However, in this case, the construction and study of convergence is simpler and more transparent. The highlighted benefits become more evident in studying second-order schemes.

or

n

7.4.2 Factorized schemes Special attention should be given to two-component splitting schemes. In particular, as a major class of additive schemes with two-component splitting, we consider factorized schemes, which we have already discussed in detail. In some cases, these schemes can be treated as special variants of additive schemes constructed on the basis of an approximation of a transition operator. The factorized scheme of the second order (the Peaceman–Rachford algorithm) has a multiplicative structure of a transition operator. Concerning the model problem (7.35)–(7.37), this factorized scheme may be written in the form B

y nC1  y n C Ay n D f .t nC1=2 /, 

t n 2 ! ,

(7.75)

where

1 B˛ D E C A˛ , ˛ D 1, 2. 2 The scheme (7.75), (7.76) may be written as (7.54) with  1  1 1 1 n ' D E C A2 f .t nC1=2 /. E C A1 2 2 B D B1 B2 ,

(7.76)

214

Chapter 7 Schemes based on approximations of a transition operator

Such an approximation of the right-hand side with accuracy O. 2 / coincides with the approximation (7.74). The transition operator has the representation  1  1    1 1 1 1 S D E C A2 E C A1 E  A1 E  A2 . (7.77) 2 2 2 2 Direct calculations show that 1 S D E   .A1 C A2 / C  2 .A21 C 2A2 A1 C A22 /2 C O. 3 /, 2 i.e., the transition operator is approximated with the third order: exp.A/  S D  3 R,

R D O.1/.

It remains to formulate a stability condition. For the noncommutative operators A1 and A2 , the inequality (7.46) is not true. However, due to the choice of a space of grid functions, we can prove the stability of the scheme (7.43), (7.77). Multiply from the left the equality (7.43) by the operator D > 0. This allows us to write (7.43) in the form v nC1 D SQ v n ,

SQ D DSD 1 .

v n D Dy n ,

For this reason, the scheme (7.43) is stable in HD  D if the inequality kDSD 1 k  1 holds. As applied to the factorized scheme (7.75), (7.76), we put D D B2 . In this case, in view of (7.77), we have  1    1 1 1 1 1 Q S D E C A1 . E  A1 E  A2 E C A2 2 2 2 2 and therefore kSQ k  1. Thus, we have demonstrated the stability and convergence of the factorized scheme (7.75), (7.76). Previously, this result was presented in the form of Theorem 4.2. The main advantage of the scheme (7.54), (7.77) in comparison with the scheme y nC1 D SQ y n C  ' n ,

n D 0, 1, : : :

is that the scheme (7.54), (7.77) is a scheme of a time-evolving process (it provides the correct solution of the steady-state problem). Unfortunately, this feature of the scheme cannot be achieved for a general multicomponent splitting.

7.4.3 Inhomogeneous approximation of a transition operator In fact, we considered a class of approximations for the transition operator of type (7.51); they are associated with the use of the classical two-level schemes with weights.

215

Section 7.4 Further additive schemes

The inhomogeneity in this class of schemes (the use of various expressions for individual operator components in the multiplicative (7.44) or additive (7.59) approximations) results from a choice of distinct weight parameters ˛ , ˛ D 1, 2, : : : , p. In more general terms, it is possible to employ approximations of various types. In the framework of the concept of SM-stability, we noted that some approximations are preferable for problems with self-adjoint operators, and quite distinct approximations are needed for problems with skew-symmetric operators. In the general problem (7.35), (7.36), for obtaining unconditionally SM-stable schemes, we can apply the following decomposition of the problem operator: 1 1 A D A1 C A2  0, A1 D A1 D .A C A /, A2 D A2 D .A  A /. (7.78) 2 2 In designing an additive scheme of second-order accuracy for the problem (7.35), (7.36), (7.78), we start with the approximation (7.70). In the scheme (7.54), we set S D S1 S2 S1 , under the condition that    S1 D exp  A1 C O. 3 /, 2

S2 D exp.A2 / C O. 3 /.

(7.79)

(7.80)

A choice of the transition operators S˛ , ˛ D 1, 2 in (7.79), (7.80) is subjected to the conditions of SM-stability. For the self-adjoint part, we put 1  1 1 . (7.81) S1 D E C A1 C  2 A21 2 8 For the skew-symmetric part, we can take the operator 1    1 1 E  A2 . (7.82) S2 D E C A2 2 2 The implementation of the additive scheme (7.54), (7.79)–(7.82) involves the inversion of the operators 1 1 1 E C A1 C  2 A21 , E C A2 , 2 8 2 which have a different design.

7.4.4 Schemes of higher-order approximation The development of additive schemes of third and higher order is problematic. Here we present some results obtained in this direction for the case of two-component splitting (p D 2). It seems natural, starting with (7.70), to employ the approximation exp. .A1 C A2 // D

p Y ˛D1

exp.b˛ A2 / exp.a˛ A1 / C O. mC1 /.

(7.83)

216

Chapter 7 Schemes based on approximations of a transition operator

To ensure the stability of the approximate solution of the problem (7.1), (7.37) with the general non-negative definite operators A1 and A2 , the real coefficients a˛ , b˛ , ˛ D 1, 2, : : : , p in (7.83) must be non-negative. For m > 2, the appropriate additive schemes do not exist. More precisely, it was shown (see, e.g., [162, 175, 176]) that there exists ˛ such that a˛ < 0 or b˛ < 0 in the representation (7.83). Below, we study a special variant of additive schemes, where one of the operators in the two-component splitting (A2 in (7.78)) is skew-symmetric. That is why the coefficients b˛ , ˛ D 1, 2, : : : , p may be non-negative, i.e., only a˛ , ˛ D 1, 2, : : : , p must be non-negative. In the papers [15, 48], it was found that for the additive schemes based on (7.83) with m > 2, there exist ˛ and ˇ such that a˛ < 0 and bˇ < 0. We cannot overcome the limit of the second order for additive schemes (see [15, 162]) using constructions more general than (7.83) p p Y Y c˛ exp.b˛ˇ A2 / exp.a˛ˇ A1 / C O. mC1 /. exp. .A1 C A2 // D ˛D1

ˇ D1

In the literature (see, e.g., [160]), the possibility of constructing additive schemes of high-order approximation using the Richardson extrapolation is discussed. By choosing the second-order approximation (7.70) as the base, for the fourth-order approximation, we have     4   1 ˆ  ˆ. / C O. 5 /, exp. .A1 C A2 // D ˆ 3 2 2 3 where       ˆ. / D exp  A1 exp.A2 / exp  A1 . 2 2 A more complicated structure for schemes of the fourth order is based on the approximation       45    ˆ ˆ exp. .A1 C A2 // D ˆ 64 3 3 3       13 1 ˆ  ˆ. / C O. 5 / C ˆ 2 2 2 64 and uses calculation results obtained with three values of a time step. The application of such extrapolation procedures leads to the loss of absolute stability of additive schemes and has no practical sense. In fact, the only way to obtain additive schemes of higher order is the use of complex coefficients ˛i , ˇi , i D 1, 2, : : : , p in (7.83). Various classes of such schemes with a non-negative real part of the coefficients were proposed [11,43,176]. Here is a typical result from [21]. We have the representation exp. .A1 C A2 // D exp.ˇ1 A2 / exp.a1 A1 / exp.b2 A2 / exp.a2 A1 /

exp.b3 A2 / exp.a2 A1 / exp.b2 A2 /

exp.a1 A1 / exp.b1 A2 / C O. 5 /,

(7.84)

Section 7.4 Further additive schemes

with

217

1 1 4 4 i 2i i a1 D a2 D , b1 D  , b1 D C , b3 D  . 4 10 30 15 15 15 15 The main peculiarity of the presented approximation is that for one operator (in our case, it is A1 ) the coefficients are real. The practical use of additive schemes with complex coefficients leads to the significant complication of computational implementation in comparison with the case of real coefficients. We have to apply complex arithmetic or solve the corresponding systems for the real and imaginary parts. For this class of SM-stable schemes, the schemes of type (7.84) are ineffective due to the fact that we have an additional splitting of the problem operator over (7.37) with p D 2. This results in additional problems with tracking characteristics of an operator of the transition to a new time level.

Chapter 8

Vector additive schemes

Here we study a new class of additive difference schemes of multicomponent splitting, which are schemes of full approximation – at each time level we determine the approximate solution of the problem under consideration. The original problem is formulated as a vector problem; in this case, instead of a single approximate solution, we search for a vector of approximate solutions. The construction of vector additive schemes is conducted on a unified methodological basis using results of the general theory of stability for operator-difference schemes and the regularization principle. Two- and three-level schemes for evolutionary equations of the first and second order are investigated.

8.1 Vector schemes for first-order equations We consider difference schemes of full approximation with an additive splitting of a positive problem operator into the sum of several operators for solving the Cauchy problem of first order. The construction of unconditionally stable vector schemes is based on the regularization principle for difference schemes. Such schemes were proposed and investigated by Abrashin (see [1,2] and the review [4]). The study of vector schemes in terms of the general theory of stability for operator-difference schemes was performed in [65, 188, 193].

8.1.1 Vector differential problem Let, as usual, H be a finite-dimensional Hilbert space, where the scalar product and norm are ., / and k  k, respectively. In H , we seek the approximate solution of the Cauchy problem for the evolutionary first-order equation du C Au D f .t /, dt u.0/ D u0 .

t > 0,

(8.1) (8.2)

Assume that the linear operator A is constant (independent of t ) and non-negative in H . For this reason, the solution of the problem (8.1), (8.2) satisfies the estimate for stability with respect to the right-hand side and the initial data: Z t 0 kDu.t /k  kDu k C kDf .s/kds, (8.3) 0

219

Section 8.1 Vector schemes for first-order equations

where, e.g., D D E, A. The estimate of type (8.3) is a reference point in obtaining estimates for the corresponding difference schemes. Additive schemes for the problem (8.1), (8.2) are based on the representation AD

p X

A˛ ,

A˛  0,

˛ D 1, 2, : : : , p.

(8.4)

˛D1

In this case, the procedure of numerically solving (8.1), (8.2) involves a sequence of p problems, and each of them is characterized by the individual operator A˛ , ˛ D 1, 2, : : : , p. For p D 2, we have various classes of unconditionally stable schemes: ADI methods, factorized schemes, predictor-corrector techniques and so on. For p > 2, the well-known unconditionally stable schemes are based on the method of summarized approximation. Instead of finding the scalar function u, we seek a vector u D ¹u1 , u2 , : : : , up º. Each individual component is defined as the solution of similar problems p X du˛ Aˇ uˇ D f .t /, C dt

t > 0,

(8.5)

ˇ D1

u˛ .0/ D u0 ,

˛ D 1, 2, : : : , p.

(8.6)

In this case, u˛ .t / D u.t /, ˛ D 1, 2, : : : , p, and therefore we can treat any component of the vector u.t / as the solution to the original problem (8.1), (8.2), (8.4). Rewrite the Cauchy problem for the system of equations as a single equation of first order for the vector unknowns. Let us also perform a preliminary transformation of the system of equations (8.5). Define the vector of the right-hand sides F .t / D ¹A1 f .t /, A2 f .t /, : : : , Ap f .t /º and the diagonal operator matrix B D ¹A˛ ı˛ˇ º, where ı˛ˇ is Kronecker’s symbol. Multiplying the individual equations (8.5) from the left by A˛ , ˛ D 1, 2, : : : , p, we get the system of equations B

du C Au D F .t /, dt

t > 0.

(8.7)

For the operator matrix A, we have the representation A D ¹A˛ Aˇ º. In view of (8.6), the system of equations (8.7) is supplemented with the initial conditions u.0/ D u0  ¹u0 , u0 , : : : , u0 º.

(8.8)

It is natural to treat the problem (8.7), (8.8) in a vector Hilbert space H D H p , where the scalar product is given by .u, v/ D

p X ˛D1

.u˛ , v˛ /.

220

Chapter 8 Vector additive schemes

In view of the equality .A˛ Aˇ / D Aˇ A˛ , the operator A is self-adjoint. In addition, we verify directly that  X 2  p u˛ , 1 . (8.9) .Au, u/ D ˛D1

It follows from (8.9) that the operator A is non-negative. In the splitting (8.4), the operator B is also non-negative. We obtain the vector problem (8.7), (8.8) with B  0,

A D A   0.

(8.10)

For the problem (8.7), (8.8), (8.10), vector difference schemes are constructed using the general theory of stability for operator-difference schemes. The feature of the problem is the non-negativity of the operators B and A; recall that the main results were obtained for positive operators. That is why we need to conduct the corresponding manipulations separately in each particular case.

8.1.2 Stability of vector additive schemes Introduce the usual uniform time grid with a step  > 0. The approximate solution of the vector problem (8.7), (8.8) at the time level t D t n is denoted by y n D ¹y1n , y2n , : : : , ypn º. We start with studying stability of vector schemes with respect to the initial data. The corresponding two-level difference schemes may be written in the canonical form y nC1  y n (8.11) C Ay n D 0, n D 0, 1, : : : . B  The investigation of stability is based on the following statement. Lemma 8.1. If, in the scheme .8.11/, the operator A is self-adjoint and constant, then under the condition of fulfillment of the operator inequality B

 A 2

(8.12)

the inequality .Ay nC1 , y nC1 /  .Ay n , y n /

(8.13)

holds. Proof. To obtain the above estimate, it is is sufficient to reproduce the proof of Theorem 2.2. It is natural to try to interpret the inequality (8.13) applied to the scheme for the problem (8.7), (8.8) as an estimate for stability with respect to the initial data. For the scheme (8.11), it is true in the case of the additional assumption on the positiveness of the operator A.

221

Section 8.1 Vector schemes for first-order equations

The creation of stable schemes for the problem (8.7), (8.8) is conducted on a unified methodological basis – the regularization principle for difference schemes. The regularization theory for difference schemes is treated as the principle of improving the quality of a difference scheme by means of introducing new additional terms (regularizers) in operators of the original scheme. Designing stable difference schemes via the regularization principle is implemented in the following way. For the original (generating) problem, we consider some simple scheme that does not have the necessary stability properties. Next, this scheme is written in the canonical form and stability restrictions of the scheme are made weaker through perturbations of the scheme operators. For the problem (8.7), (8.8) with F .t / D 0, it seems reasonable to take the explicit scheme y nC1  y n (8.14) C Ay n D 0, n D 0, 1, : : : B  as the generating scheme. By (8.9), the inequality (8.12) for the scheme (8.14) is equivalent to the inequality  X 2  p p X   .A˛ y˛ , y˛ /  A˛ y˛ , 1  0. (8.15) .By, y/  .Ay, y/ D 2 2 ˛D1 ˛D1 The following inequality is satisfied: 2 X p p X A˛ y˛  p .A˛ y˛ /2 . ˛D1

(8.16)

˛D1

In the explicit schemes, restrictions on a time step are associated with the upper estimates for discrete operators. For non-self-adjoint operators A˛ , ˛ D 1, 2, : : : , p, we assume that the inequalities 1 kA˛ y˛ k2 , ˛ D 1, 2, : : : , p (8.17) .A˛ y˛ , y˛ / 

˛ are valid. In view of (8.16), (8.17), the inequality (8.15) can be transformed as  X X 2   p p p X  1 p .A˛ y˛ , y˛ /  A˛ y˛ , 1  kA˛ y˛ k2  0,   2

2 ˛D1 ˛D1 ˛D1 where D max ˛ . It is true for ˛



2 . p

(8.18)

This restriction on a time step may be treated as a generalization of the standard condition for stability of explicit schemes for the problem (8.1), (8.2) to the case of explicit difference schemes for the vector problem (8.7), (8.8).

222

Chapter 8 Vector additive schemes

If (8.18) is valid, then, by Lemma 8.1, for the scheme (8.14), the a priori estimate .Ay nC1 , y nC1 /  .Ay n , y n /,

n D 0, 1, : : :

(8.19)

holds. The scheme (8.14) is constructed to determine the solution of the original problem (8.1), (8.2). The individual components of the vector y have no particular sense; the approximate solution is the function yN that is defined as follows: AyN D

p X

A˛ y˛ .

(8.20)

˛D1

Taking into account (8.4), we can rewrite (8.19) in the form kAyN nC1 k  kAyN n k.

(8.21)

By the positiveness of the operator A, the inequality (8.21) is the desired estimate for stability of the approximate solution of (8.1), (8.2) with respect to the initial data in the case of using the explicit scheme (8.12). Taking into account the condition (8.12), it is natural to construct a class of regularized difference schemes for the problem (8.7), (8.8), (8.10) on the basis of a perturbation of the operator B in the scheme (8.14). Consider the scheme (8.11) with B D B C R,

A D A,

(8.22)

where  is a regularization (perturbation) parameter and R is a regularizer. Remaining in the class of diagonal operator matrices B, we set ¯ ® (8.23) R D A˛ A˛ ı˛ˇ . Theorem 8.1. For the scheme .8.11/, .8.22/, .8.23/ with   1 p  ,  2 

(8.24)

the estimate .8.20/, .8.21/ holds. Proof. For the scheme (8.11), (8.22), (8.23), the inequality (8.12) can be transformed, similarly to (8.15)–(8.17), as  X 2  p p p X X   2 .A˛ y˛ , y˛ / C  kA˛ y˛ k  A˛ y˛ , 1 .By, y/  .Ay, y/ D 2 2 ˛D1 ˛D1 ˛D1  X p 1 p  kA˛ y˛ k2  0. C 

2 ˛D1 This inequality is valid under the constraint (8.24) on the regularization parameter . From (8.13), it follows the stability estimate (8.21) for the approximate solution of the problem (8.1), (8.2). The estimate (8.24) means that the schemes with   0.5p will be unconditionally stable.

223

Section 8.1 Vector schemes for first-order equations

The regularized scheme (8.11), (8.22), (8.23) results from the following scheme with weights for the system of equations (8.5) with  D   : p X y˛nC1  y˛n Aˇ yˇn D ' n , C .E C  A˛ /  ˇ D1

(8.25)

˛ D 1, 2, : : : , p. The transition to a new time level in the scheme (8.25), as well as in the standard (scalar) version of additive schemes, is attributed to the inversion of the operators EC A˛ , ˛ D 1, 2, : : : , p at every time step. It should be noted that the scheme (8.25) for problems with p > 2 has some peculiarity if the weight  is selected larger than unity. The essential point is the requirement to store the whole vector that may impose additional costs on computer memory. The same remark is true for some (scalar) variants of additive difference schemes, e.g., for additively averaged difference schemes.

8.1.3 Stability with respect to the right-hand side The study on convergence of difference schemes for unsteady problems is based on estimates for stability of the difference solution with respect to the right-hand side. In many cases, we can focus on elementary estimates with respect to the right-hand side that are directly derived from estimates for stability with respect to the initial data. Peculiarities of the above vector schemes are associated with very weak restrictions (8.10) on the operators of the problem (8.7), (8.8); this requires a separate investigation. We analyze stability of the schemes of type (8.25) with various right-hand sides. This situation occurs if we study the problem for the error. Let .E C  A˛ /

p X y˛nC1  y˛n Aˇ yˇn D '˛n , C  ˇ D1

(8.26)

˛ D 1, 2, : : : , p. Suppose B˛ D E C  A˛ , ˛ D 1, 2, : : : , p, and so B˛ > 0; therefore, B˛1 does exist. The scheme (8.26) may be rewritten as y˛nC1 D

p X

s˛ˇ y˛n C B˛1 '˛n ,

˛ D 1, 2, : : : , p

ˇ D1

with

s˛ˇ D ı˛ˇ  B˛1 Aˇ .

Multiplying by A˛ and adding these equations, we obtain p X ˛D1

A˛ y˛nC1 D

p p X X ˛D1 ˇ D1

A˛ s˛ˇ y˛n C 

p X ˛D1

A˛ B˛1 '˛n .

(8.27)

224

Chapter 8 Vector additive schemes

In view of (8.20) and the above-proved stability with respect to the initial data under the restriction (8.24), from (8.27), we have kAyN nC1 k  kAyN n k C 

p X

kA˛ B˛1 '˛n k.

(8.28)

˛D1

Thus, for the difference scheme (8.26) under the condition (8.24), we obtain the a priori estimate with respect to the initial data and the right-hand side. In comparison with (8.28), we have the simpler estimate kAyN nC1 k  kAyN n k C 

p X

kA˛ '˛n k.

˛D1

Using (8.28), we can prove for the error problem the convergence of the vector additive scheme with the first order in time. Other variants of vector additive schemes are discussed below.

8.2 Stability of vector additive schemes in Banach spaces Consider the problems of constructing unconditionally stable difference schemes in finite-dimensional Banach spaces. As in the case of the classical additive schemes of componentwise splitting, the stability property is a consequence of the stability of discrete problems for the individual operator terms.

8.2.1 Problem formulation The most complete results on stability of additive schemes were obtained in the study of difference schemes in Hilbert spaces of grid functions on the basis of the general theory of stability (well-posedness) for operator-difference schemes. This is also true for stability and convergence of vector additive operator-difference schemes of full approximation. In many problems, the fundamental issue is the well-posedness of a scheme in the uniform norm (in the Banach space of grid functions L1 ). Here, according to the work [158], we study stability of the simplest vector additive scheme in arbitrary norms. The stability of the additive scheme is shown under the condition that the fully implicit schemes for the individual components are stable. Consider the Cauchy problem for the system of ODEs of first order: X dui aij .t /uj D fi .t /, C dt m

t > 0,

i D 1, 2, : : : , m.

(8.29)

j D1

The system (8.29) occurs as a result of discretization in space considering, e.g., initialboundary value problems for parabolic equations.

Section 8.2 Stability of vector additive schemes in Banach spaces

225

Setting u D u.t / D ¹u1 , u2 , : : : , um º, A D ¹aij º, we rewrite (8.29) in the matrix (operator) form du C A.t /u D f .t /. (8.30) dt The Cauchy problem (8.30) is considered for t > 0 with the initial data u.0/ D u0 ,

u0 D ¹u01 , u02 , : : : , u0m º.

(8.31)

Additive schemes for numerically solving the problem (8.30), (8.31) are based on the representation AD

p X

A˛ ,

A˛  0,

˛ D 1, 2, : : : , p.

(8.32)

˛D1

For simplicity, we restrict ourselves to problems where each operator term A˛ , ˛ D 1, 2, : : : , p is a constant (independent of time) operator. We define a class of additive schemes as follows. Introduce a uniform time grid with a step  > 0 and let y n be the approximate solution at the time moment t n D n , n D 0, 1, : : : . Assume that the fully implicit difference schemes y˛nC1  y˛n C A˛ y˛nC1 D '˛nC1 , n D 0, 1, : : : , ˛ D 1, 2, : : : , p 

(8.33)

are unconditionally stable in a Banach space with a norm k  k. More precisely, we suppose that the following levelwise estimates hold for the difference schemes (8.33): ky˛nC1 k  ky˛n k C  k'˛nC1 k, n D 0, 1, : : : , ˛ D 1, 2, : : : , p.

(8.34)

The above class of additive schemes include schemes with non-negative definite operators (A˛  0, ˛ D 1, 2, : : : , p in the Hilbert space of grid functions L2 ). Another important example is a difference scheme for problems with nonstrict diagonal dominance by rows for each operator term M X

ai i 

jaij j,

i D 1, 2, : : : , M .

i6Dj D1

In this case, the scheme (8.33) is stable in L1 .

8.2.2 Vector additive scheme To solve numerically (8.30)–(8.32), we use (see [1]) the two-level vector additive scheme p ˛ X X y˛nC1  y˛n nC1 Aˇ yˇ C Aˇ yˇn D ' n , C  ˇ D1

n D 0, 1, : : : ,

ˇ D˛C1

˛ D 1, 2, : : : , p.

(8.35)

226

Chapter 8 Vector additive schemes

The initial data, in accordance with (8.31), are given in the form y˛0 D u0 ,

˛ D 1, 2, : : : , p.

(8.36)

We have already examined in detail the scheme X y nC1  y˛n Aˇ yˇn D ' n , .E C  A˛ / ˛ C  p

˛ D 1, 2, : : : , p,

ˇ D1

which was constructed on the basis of regularization (perturbation) using a diagonal matrix. In this case, we obtain asynchronous algorithms, where the individual elements y˛ are evaluated at a new time level independently from each other. In this sense, they are similar to the additively averaged schemes of componentwise splitting. In the scheme (8.35), (8.36), similarly to the transition from the iterative Jacobi method to the Gauss–Seidel technique, the solution component that is already determined at the new time level is involved in the calculation of the following components of the solution vector. In this case, we may obtain advantages that are typical for the fully implicit difference schemes. We prove the unconditional stability of the vector additive scheme (8.35), (8.36).

8.2.3 Study on stability Considering the equations for y1nC1 and ypn , from (8.35), we obtain directly the equality .E C A1 /

ypn  ypn1 y1nC1  y1n ' n  ' n1 D C .   

(8.37)

nC1 , we get the second useful Subtracting the equation for y˛nC1 from the equation for y˛C1 relation n y nC1  y˛C1 y nC1  y˛n D ˛ , .E C A˛C1 / ˛C1   (8.38)

n D 0, 1, : : : ,

˛ D 1, 2, : : : , p  1.

From (8.38), in view of (8.33), (8.34), it follows immediately that     y nC1  y n    y˛nC1  y˛n   ˛C1 ˛C1  ,           n D 0, 1, : : : ,

(8.39)

˛ D 1, 2, : : : , p  1.

Similarly, from (8.37), we obtain      n  nC1 n1    y1  y1n   ypn  ypn1    C  '  ' .          

(8.40)

Section 8.2 Stability of vector additive schemes in Banach spaces

227

By (8.39), from (8.40), we get the levelwise estimate for the time derivative of the approximate solution:      n  nC1 n1    y˛  y˛n   y˛n  y˛n1    C  '  ' ,           (8.41) n D 1, 2, : : : ,

˛ D 1, 2, : : : , p.

From (8.41) follows the inequality:    nC1    n  ' k  ' k1   y˛  y˛n   y˛1  y˛0  X     C  ,          kD1

n D 1, 2, : : : ,

(8.42)

˛ D 1, 2, : : : , p.

From (8.35) with ˛ D 1, taking into account (8.32) and (8.36), we have the inequality   1  y1  y10    k' 0  Au0 k.     By (8.39), the inequality (8.42) may be written in the form    nC1  n  ' k  ' k1  X  y˛  y˛n       k' 0  Au0 k C  ,       kD1

n D 1, 2, : : : ,

(8.43)

˛ D 1, 2, : : : , p.

Taking into account the obvious inequality ky˛nC1 k



ky˛n k

 nC1   y˛  y˛n   , C  

from (8.43), we obtain the desired estimate for each individual component of the vector additive scheme (8.35), (8.36):   n  ' k  ' k1  X   ky˛nC1 k  ky˛n k C  k' 0  Au0 k C   ,    (8.44) kD1

n D 1, 2, : : : ,

˛ D 1, 2, : : : , p.

This allows us to formulate the following main result. Theorem 8.2. Assume that the implicit difference schemes .8.33/ satisfy the a priori estimates .8.34/. Then the vector additive scheme .8.32/, .8.35/, .8.36/ is stable with respect to the initial data and the right-hand side, and the approximate solutions satisfy the estimates .8.44/.

228

Chapter 8 Vector additive schemes

We emphasize that the estimates obtained above are the estimates for stability for each individual component y˛n , ˛ D 1, 2, : : : , p. Thus, we can treat any component or their linear combination as an approximate solution of the problem (8.29), (8.30): n

y D

p X

p X

c˛ y˛n ,

˛D1

c˛ D 1,

c˛  0,

˛ D 1, 2, : : : , p.

˛D1

Using the estimates of type (8.44), convergence of vector additive schemes is investigated by standard methods. In this regard, we highlight that the estimate (8.44) is very close to the basic a priori estimates for the two-level difference schemes, which are based on time derivatives of the right-hand side. This estimate for stability with respect to the initial data and the right-hand side for the vector difference scheme employs similar estimates for the schemes (8.33), which include individual terms of the operator splitting (8.32) only. Such problems were already studied in the stability theory for operator-difference schemes [131,134]. In particular, the exact necessary and sufficient condition was obtained, convergence was investigated in various norms with minimal requirements on problem operators and so on.

8.3 Schemes of second-order accuracy We consider the vector additive schemes of multicomponent splitting that converge with the second order by  . Using the regularization principle, three-level difference schemes are constructed; the class of alternating triangle vector additive schemes is highlighted.

8.3.1 Statement of the problem We consider the model problem du C Au D f .t /, dt

t > 0,

u.0/ D u0 , where AD

p X

A˛ ,

A˛ D A˛  0,

(8.45) (8.46)

˛ D 1, 2, : : : , p

(8.47)

˛D1

and all operators A˛ , ˛ D 1, 2, : : : , p are constant. Possible generalizations of the results under consideration to a more general case of non-self-adjoint time-dependent operators is not discussed here.

229

Section 8.3 Schemes of second-order accuracy

From (8.45)–(8.47), we pass to the vector problem D

du C Au D F .t /, dt

t > 0,

(8.48)

u.0/ D u0  ¹u0 , u0 , : : : , u0 º.

(8.49)

For the operator matrix D, A, we have the representation ¯ ® D D D  D A˛ ı˛ˇ  0, ¯ ® A D A  D A˛ Aˇ  0,

(8.50) (8.51)

and the right-hand side of equation (8.48) is ® ¯ F .t / D A1 f .t /, A2 f .t /, : : : , Ap f .t / . For the numerical solution of the vector problem (8.48), (8.49), using the stability theory for operator-difference schemes, we have already considered two-level difference schemes of first-order accuracy. A similar study can be conducted in constructing schemes of second-order accuracy.

8.3.2 Three-level vector schemes The condition for stability of three-level operator-difference schemes provides the basis of the present study. To analyze stability of three-level vector schemes, we use the canonical form B

y nC1  y n1 C R.y nC1  2y n C y n1 / C Ay n D 0, 

(8.52)

n D 1, 2, : : : . The following statement is valid (see the proof of Theorem 2.10). Lemma 8.2. Assume that in .8.52/, the operators A and R are constant and selfadjoint. Then under the condition B0 (8.53) the inequality E nC1  E n

(8.54)

holds; we employ the following notation: 1 E nC1 D .A.y nC1 C y n /, y nC1 C y n / C .R.y nC1  y n /, y nC1  y n / 4 (8.55) 1  .A.y nC1  y n /, y nC1  y n /. 4

230

Chapter 8 Vector additive schemes

We focus on the schemes where E nC1 defines the appropriate norm of the solution. If A > 0, then this implies that the following inequalities are valid: B  0,

4R  A  0,

(8.56)

which present the necessary and sufficient conditions for stability of the three-level scheme (8.52) (see [131, 134], Theorem 2.10). As in the case of the two-level vector scheme, the solution is the function yN that is defined as follows: p X A˛ y˛ . AyN D ˛D1

Therefore, for the three-level scheme (8.52) under the condition (8.56), the value E nC1 defines the norm, although the positiveness of the operator A, in general, is not satisfied. Unconditionally stable three-level vector additive schemes are constructed using the regularization principle. For the homogeneous problem with F .t / D 0, we can take the simplest explicit scheme of the second-order approximation as the generating scheme: y nC1  y n1 (8.57) C Ay n D 0, n D 1, 2, : : : B 2 In accordance with (8.56), this scheme is classified as an absolutely unstable scheme. Taking into account the necessary and sufficient conditions (8.56), we create absolutely stable regularized schemes through perturbations of the operator R. To preserve the second order by  , we put B D D,

R D C

(8.58)

¯ ® C D A˛ A˛ ı˛ˇ .

(8.59)

with the above-used regularizer

Theorem 8.3. The scheme .8.52/, .8.58/, .8.59/ is absolutely stable with respect to the initial data for   p=4, and the approximate solution satisfies the estimate .8.54/, .8.55/. Proof. For this scheme, the first condition in (8.56) is valid, and the second condition takes the form  X 2  p p X 2 kA˛ y˛ k  A˛ y˛ , 1 4.Ry, y/  .Ay, y/ D 4 ˛D1

 .4  p/

˛D1 p X

kA˛ y˛ k2  0.

˛D1

This implies the desired restriction on the parameter .

231

Section 8.3 Schemes of second-order accuracy

In accordance with general results of the stability (well-posedness) theory for threelevel operator-difference schemes [131, 134], it is possible to obtain estimates for stability with respect to the initial data and the right-hand side that allow to study completely the convergence of vector additive schemes discussed here.

8.3.3 Schemes of the alternating triangle method Among the most interesting schemes of two-component splitting, we emphasized additive difference schemes of the alternating triangle method. They are based on the splitting of a self-adjoint problem operator into the sum of two operators that are adjoint to each other. Here this class of schemes is applied to designing vector additive schemes with second-order accuracy. Consider the problem (8.48), (8.49). For the problem operator, we employ the decomposition (8.60) A D A1 C A2 , A1 D A2 . By (8.51), for the individual components of this splitting, we have .˛/

A˛ D ¹Aij º,

˛ D 1, 2, 8 8 i < j, Ai Aj , 0, i < j, ˆ ˆ ˆ ˆ ˆ ˆ < 1 < 1 .1/ .2/ Ai Ai , i D j , Ai Ai , i D j , Aij D Aij D ˆ ˆ 2 2 ˆ ˆ ˆ ˆ : : i > j. 0, i > j, Ai Aj , Thus, the operator matrix is split into a lower triangular and an upper triangular matrix. In this interpretation, we can speak of a block operator variant of the alternating triangle method. An additive scheme for the problem (8.50), (8.51), (8.60) corresponds to the use of D

y nC1=2  y n C A1 y nC1=2 C A2 y n D F nC1=2 , 0.5

(8.61)

y nC1  y nC1=2 (8.62) C A1 y nC1=2 C A2 y nC1 D F nC1=2 . 0.5 The implementation of the vector additive schemes (8.61), (8.62) with the above splitting involves the alternating inversion of the upper and lower triangular operator matrices. For example, the first stage (equation (8.61)) corresponds to the solution of the p problems D

X  y˛n y˛ nC1=2 Aˇ yˇ C A˛ C 0.5

nC1=2

y˛

˛1

ˇ D1

n D 0, 1, : : : ,

˛ D 1, 2, : : : , p.

nC1=2

2

C y˛n

C

p X ˇ D˛C1

Aˇ yˇn D ' nC1=2 ,

232

Chapter 8 Vector additive schemes

For the second stage (equation (8.62)), we get nC1=2

˛1 X

y˛nC1  y˛ 0.5

C

n D 0, 1, : : : ,

˛ D 1, 2, : : : , p.

ˇ D1

nC1=2 Aˇ yˇ

nC1=2

y nC1 C y˛ C A˛ ˛ 2

C

p X ˇ D˛C1

Aˇ yˇnC1 D ' nC1=2 ,

The additive scheme (8.61), (8.62) may be written in the canonical form (with F .t / D 0) as the two-level vector scheme B

unC1  un C Aun D 0, 

n D 0, 1, : : : .

For positive A˛ , ˛ D 1, 2, : : : , p in the splitting (8.47), the operator B has the factorized form       1 B D D C A1 D D C A2 . (8.63) 2 2 The further study is performed by the standard procedure. In the case of (8.63), we have  2 B D B  D D C A C A1 D 1 A2 . 2 4 The stability condition  B A 2 holds, and therefore the vector additive scheme of the alternating triangle method (8.61), (8.62) is unconditionally stable; it has second-order approximation by  for the problem (8.45)–(8.47). It is easy to obtain the appropriate estimates for stability with respect to the initial data and the right-hand side in various norms.

8.4 Vector schemes for equations of second order We construct vector additive schemes (multicomponent alternating direction schemes) for numerically solving the Cauchy problem for an evolutionary equation of second order. These schemes are the schemes of full approximation – each intermediate problem approximates the original problem. The advantages of vector additive schemes, in particular, consist in the fact that splitting schemes for evolutionary equations of second order are constructed practically in the same way as for the first-order equations. Such schemes were considered in many works; we should mention [1, 2]. In our research, we generalize the results of the paper [3].

8.4.1 The Cauchy problem for a second-order equation Consider a function y from a finite-dimensional real Hilbert space H , where the scalar product and norm are denoted by ., / and kyk D .y, y/1=2 , respectively. For an

Section 8.4 Vector schemes for equations of second order

233

operator D D D  > 0, the space HD stands for the space H equipped with the scalar product .y, w/D D .Dy, w/ and the norm kykD D .Dy, y/1=2 . We seek the solution u.t / 2 H of the Cauchy problem for the evolutionary secondorder equation d 2u C Au D f .t /, dt 2

t > 0,

u.0/ D u0 ,

(8.64) (8.65)

du (8.66) .0/ D v 0 . dt We restrict ourselves to the simplest case of the positive, self-adjoint, and timeindependent operator A. Recall the stability estimate with respect to the initial data and the right-hand side for the problem (8.64)–(8.66), which can be obtained by multiplying scalarly in H equation (8.64) by u. Lemma 8.3. For the problem .8.64/–.8.66/, the a priori estimate Z t kf .s/kds ku.0/k  ku0 kA C kv 0 k C

(8.67)

0

is valid where ku.0/k2



2 kukA

 2  du   C  dt  .

This estimate serves as a checkpoint in the construction of additive schemes for the problem (8.64)–(8.66). Assume that the operator A may be represented as AD

p X

A˛ ,

A˛ D A˛  0,

˛ D 1, 2, : : : , p.

(8.68)

˛D1

Additive difference schemes are based on the representation (8.68), where the transition from one time level t n to the next level t nC1 D t n C  (here  > 0 is a time step) is associated with solving problems for the individual constant operators A˛ , ˛ D 1, 2, : : : , p in the additive decomposition (8.68). Thus, the original problem is divided into p simpler subproblems.

234

Chapter 8 Vector additive schemes

8.4.2 Vector problem We introduce the vector u D ¹u1 , u2 , : : : , up º. Each individual component is defined as the solution of the one-type problems p X d 2 u˛ C Aˇ uˇ D f .t /, dt 2

t > 0,

(8.69)

ˇ D1

u˛ .0/ D u0 ,

(8.70) du˛ (8.71) .0/ D v 0 , ˛ D 1, 2, : : : , p. dt It is clear that u˛ .t / D u.t /, and therefore any component of u.t / may be treated as the solution of the original problem (8.64)–(8.66). We perform a preliminary transformation of the system of equations (8.69). Multiplying equation (8.69) from the left by A˛ , ˛ D 1, 2, : : : , p, we obtain the system of equations d 2u (8.72) D 2 C Au D F .t /, t > 0. dt ® ¯ For the elements of the operator matrix A, we have A  A˛ Aˇ . For the righthand side vector, the ¯ representation F .t / D ¹A1 f .t /, A2 /f .t /, : : : , Ap f .t /º is true, ® and D  A˛ ı˛ˇ is a diagonal operator matrix, where ı˛ˇ stands for Kronecker’s symbol. In view of (8.70), (8.71), the system of equations (8.72) is supplemented with the initial data u.0/ D u0  ¹u0 , u0 , : : : , u0 º,

(8.73)

du (8.74) .0/ D v0  ¹v 0 , v 0 , : : : , v 0 º. dt It is natural to study the problem (8.72)–(8.74) in the vector Hilbert space H p , where the scalar product is given by .u, v/ D

p X

.u˛ , v˛ /.

˛D1

By self-adjointness of the operators A˛ , ˛ D 1, 2, : : : , p, the operator A in H p is self-adjoint, too. In addition, we obtain immediately that  X 2  p A˛ u ˛ , 1 . (8.75) .Au, u/ D ˛D1

From (8.75), it follows directly that the operator A is non-negative. In the splitting (8.68), the operator D is also non-negative and self-adjoint. We get the vector problem (8.72)–(8.74), where (8.76) D D D   0, A D A   0.

235

Section 8.4 Vector schemes for equations of second order

The specific feature of the above vector problem consists in the non-negativity of the operators D and A that does not allow us to directly use general results of the stability theory for three-level difference schemes [131, 134].

8.4.3 Scheme with weights Let us present an estimate for stability with respect to the initial data and the righthand side of the standard scheme with weights for the problem (8.64)–(8.66). Apply the second-order scheme y nC1  2y n C y n1 C A.y nC1 C .1  2 /y n C y n1 / D f n , 2

(8.77)

where y 0 , y 1 is given. An analog of Lemma 8.3 is the following statement. Lemma 8.4. The scheme .8.77/ satisfies the a priori estimate ky nC1 k  ky n k C  kf n k, where ky nC1 k2

 nC1 2 y  yn      

E C. 14 / 2 A

(8.78)

 nC1 2 y C yn    . C  2 A

Proof. The estimate (8.78) is consistent with the estimate (8.67) for the solution of the differential problem, and, for   0.25, it provides unconditional stability of the scheme with weights (8.77) with respect to the initial data and the right-hand side. To prove the lemma, it is convenient to introduce the new functions vn D

y n C y n1 , 2

wn D

y n  y n1 , 

and then

v nC1 C v n  2 w nC1  w n  . 2 4  In this new notation, the scheme (8.77) may be written in the form yn D

v nC1 C v n w nC1  w n CA D f n , n D 1, 2, : : : ,  2 where   1 2 BDEC   A. 4 Multiply scalarly equation (8.79) by B

 .w nC1 C w n / D 2.v nC1  v n /, and this yields 2 2 2 2  kw n kB C kv nC1 kA  kv n kA D  .f n , w nC1 C w n /. kw nC1 kB

(8.79)

236

Chapter 8 Vector additive schemes

For the left-hand side of this equation, we have 2 2 2 2 kw nC1 kB  kw n kB C kv nC1 kA  kv n kA

D .ky nC1 k  ky n k /.ky nC1 k C ky n k /. For the right-hand side, we get .f n , w nC1 C w n /  kf n k.kw nC1 k C kw n k/  kf n k.ky nC1 k C ky n k /. This implies the desired a priori estimate (8.78), which provides the basis for the study of convergence and accuracy of difference schemes.

8.4.4 Additive schemes To obtain a scheme with given quality, as usual, we focus on the use of the common methodological principle – the regularization principle for difference schemes. In the construction of vector additive schemes, it is reasonable to start with the explicit scheme, which is treated as the generating scheme. For the problem (8.72)– (8.74), this scheme has the canonical form B

y nC1  y n1 C R.y nC1  2y n C y n1 / C Ay n D F n 

(8.80)

with

1 D. 2 The scheme (8.80), (8.81) results from the explicit scheme B D 0,

RD

p X y˛nC1  2y˛n C y˛n1 C Aˇ yˇn D f n , 2

(8.81)

n D 1, 2, : : : .

ˇ D1

for the problem (8.69)–(8.71). Using notation y n C y n1 y n  y n1 , wn D , 2  the vector scheme, written in the canonical form (8.80), (8.81), obtains the form vn D

B

vnC1 C vn wnC1  wn CA D F n,  2

where

n D 1, 2, : : : ,

(8.82)

1 B D D   2 A. 4

Under the constraint B  0, i.e., 1 D   2 A, 4

(8.83)

237

Section 8.4 Vector schemes for equations of second order

similarly to the proof of Lemma 8.4, we get the estimate ky nC1 k  ky n k C  kF n k,

(8.84)

assuming  nC1 2 y  yn   ky nC1 k2     

 nC1 2 y C yn   . C   2 D 1  2 A A 4

To specify the condition (8.83), we employ the upper estimates for the operator terms A˛ , ˛ D 1, 2, : : : , p. Let .A˛ u˛ , u˛ /  ˛ ku˛ k2 ,

˛ D 1, 2, : : : , p

and assume that D max ˛ . For non-negative self-adjoint operators A˛ , ˛ D ˛ 1, 2, : : : , p, we use the estimates 1 kA˛ u˛ k2 , ˛ D 1, 2, : : : , p.

In view of (8.75), the inequality (8.83) may be rewritten in the form  X 2  p p X 1 2 1 2 .Du, u/   .Au, u/ D .A˛ u˛ , u˛ /   A˛ u˛ , 1  0. (8.85) 4 4 ˛D1 ˛D1 .A˛ u˛ , u˛ / 

Taking into account the inequality 2 X p p X A˛ u ˛  p .A˛ u˛ /2 , ˛D1

˛D1

from (8.85), we get p X ˛D1

.A˛ u˛ , u˛ / 

 X p ˛D1

A˛ u ˛

2   X p 1 p ,1  kA˛ u˛ k2  0.  2

4 ˛D1

Thus, we arrive at the following restriction on a time step: 4 2  p

(8.86)

for the explicit scheme (8.81). It is interesting to compare the condition (8.86) with the standard restriction on a time step for the explicit scheme for the problem (8.64)– (8.66). In accordance with the regularization principle, to remove restrictions on a time step we perturb the operator of the difference scheme. Remaining in the class of diagonal operator matrices and preserving the second-order approximation, we apply in (8.80), (8.81) the representation ¯ ® (8.87) D D .A˛ C A2˛ /ı˛ˇ .

238

Chapter 8 Vector additive schemes

This regularization corresponds to using, instead of the explicit scheme (8.81), the scheme with weights p X y˛nC1  2y˛n C y˛n1 C Aˇ yˇn C A˛ .y˛nC1 2 (8.88) ˛¤ˇ D1 C .1  2 /y˛n C y˛n1 / D f n ,

n D 1, 2, : : :

The implementation of the regularization scheme (8.88) is completely with  D analogous to the implementation of the standard splitting schemes and is associated with the inversion of operators .E C   2 A˛ /, ˛ D 1, 2, : : : , p. For the scheme (8.80), (8.87), the condition (8.83), similarly to the explicit scheme (8.81), gives     p 1 p 2 X 1 kA˛ u˛ k2  0. C   .Du, u/   2 .Au, u/  4

4 ˛D1   2.

Therefore, the estimate (8.84) for the scheme (8.80), (8.87) is valid under the restriction 1 p . (8.89)   4  2 For the stronger restriction   0.25p, the estimate (8.84) holds for any time step  , i.e., the estimate is unconditional. This establishes the following result. Theorem 8.4. For the regularized vector difference scheme .8.88/, the a priori estimate .8.84/ holds under the constraint .8.89/. Among other unconditionally stable vector additive schemes for the equation of second order, we highlight the alternating triangle method, where, in addition to (8.68), the following splitting into two triangular operator matrices: A D A1 C A2 ,

A1 D A2

is used.

8.4.5 Stability of additive schemes The a priori estimate (8.84) is the estimate for stability with respect to the initial data and the right-hand side under the restriction that kuk is a norm. In our case (see (8.76)), under the constraint (8.83), kuk is a seminorm. The passage to a norm may be associated with using the stronger, in comparison with (8.83), restriction 1 D >  2 A. 4 Without discussing various possibilities in this direction, we consider the case   0.25p, where p X 1 2 .Du, u/   .Au, u/  .A˛ u˛ , u˛ /. (8.90) 4 ˛D1

239

Section 8.4 Vector schemes for equations of second order

From (8.84), in view of (8.90), we obtain the estimate  !1=2 p  nC1 n X X  y˛  y˛n 2    ky k C  kFk k. 1     A˛ ˛D1

(8.91)

kD1

This estimate guarantees the stability of some individual component y˛ of the vector y under the additional assumption that the operator term A˛ , ˛ D 1, 2, : : : , p is positive. If A˛ > 0, ˛ D 1, 2, : : : , p, then we can consider any component y˛ , ˛ D 1, 2, : : : , p as the approximate solution of the scalar problem (8.64)–(8.66); this follows from the estimate (8.91). In the more general case, where the operator terms A˛ , ˛ D 1, 2, : : : , p are nonnegative, similarly to [193], we focus on a special definition of the approximate solution as a linear combination of the individual components. Using the specified components y˛ , ˛ D 1, 2, : : : , p, we define the scalar function yN from the relation AyN 

p X

A˛ y˛ .

(8.92)

.Ay, y/ D kAyk N 2.

(8.93)

˛D1

By (8.75), we have In view of (8.92), (8.93), from (8.84) follows the estimate kAyN nC1=2 k  ky1 k C

n X

 kFk k

(8.94)

kD1

for the approximate solution at a half-integer grid point: 1 yN nC1=2 D .yN nC1 C yN n /. 2 The estimate (8.94) may be treated as an analog of the corresponding a priori estimate for the differential problem (8.64)–(8.66); this estimate is obtained via scalar multiplication of the equation by Au.

Chapter 9

Iterative methods

In solving steady-state problems of mathematical physics, there are widely-used iterative methods that may be treated as numerical techniques in which the time-independent solution is obtained as a limit of a pseudo-time evolution process. Iterative methods are based on the determination of the approximate solutions via solving a sequence of simpler problems, which may often be associated with the use of various additive schemes. The basic results of the theory of iterative methods are presented here, and possibilities for constructing iterative methods via splitting a problem operator into the sun of simpler operators are briefly discussed. We emphasize factorized methods and highlight the iterative alternating direction method as the most popular among them. The iterative alternating triangle algorithm is recognized as one of the most efficient iterative methods. A brief analysis of the use of additive difference schemes for constructing iterative methods is also presented.

9.1 Basics of iterative methods This section describes the basic concepts of the theory of iterative methods for solving operator equations considered in finite-dimensional Hilbert spaces. Two main problems are formulated: a proper choice of iteration parameters and the construction of an operator (preconditioner) of the transition to a new iteration.

9.1.1 Problem formulation In a finite-dimensional real Hilbert space H , we seek a function y 2 H that is the solution of the operator equation Ay D '. (9.1) Here A is a positive linear operator acting in H , and ' is a given element of H . An iterative method is a method in which, starting from some initial approximation y 0 i nH , we determine, in succession, approximate solutions of equation (9.1), i.e., y 1 , y 2 , : : : , y k , : : : , , where k is the iteration number. The value y kC1 is evaluated from the previously found values y k , y k1 , : : : . If, for the calculation of y kC1 , only the value y k obtained at the previous iteration is used, then the iterative method is called a one-step (two-level) method. Accordingly, if the values of y k and y k1 are employed in this determination, then the iterative method is referred to as a two-step (three-level) method.

241

Section 9.1 Basics of iterative methods

Any two-level iterative method may be written as Bk

y kC1  y k C Ay k D ', kC1

k D 0, 1, : : :

(9.2)

According to [131, 138], this formulation is said to be the canonical form of a twolevel iterative method. For a given y 0 , all subsequent approximations to the solution are evaluated by (9.2). In this formulation of the iterative method, it is easy to see a relation between this method and difference schemes intended for the numerical solving of unsteady problems. The accuracy of an approximate solution can be adequately characterized by the error z k D y k  y. We consider convergence of iterative methods in the energy space HD generated by a self-adjoint positive definite operator D in H . In HD , the scalar product and the norm are introduced as .y, w/D D .Dy, w/,

1=2

kykD D .y, y/D ,

respectively. An iterative method converges in HD if kz k kD ! 0 as k ! 1. As a convergence measure for iterations, we use the relative error " such that at the n-th iteration (9.3) ky n  ykD  "ky 0  ykD . Due to the fact that the exact solution y is unknown, accuracy of the approximate solution is estimated by the residual r k D Ay k  ' D Ay k  Ay, which can be calculated directly. For instance, the iterative process is continued unless we obtain the estimate (9.4) kr n k  "kr 0 k. The use of the convergence criterion (9.4) implies that, in (9.3), we choose D D A A. We denote by n."/ the minimum number of iterations that guarantees the accuracy " to be achieved (the fulfillment of (9.3) or (9.4)). To construct an iterative method, we should strive to minimize the computational costs on calculating the approximate solution of the problem (9.1) with the required accuracy. Let Qk be the total number of arithmetic operations required to find the iteration y k , and suppose that n  n."/ iterations have been made. Then the computational costs can be evaluated as n X Qk . Q."/ D kD1

As for the two-level iterative method (9.2), the quantity Q./ can be minimized through a proper choice of the operators Bk and the iteration parameters kC1 . Normally, the operators Bk are considered based on some reasoning, and the iterative method (9.2) can be optimized through a proper choice of iteration parameters.

242

Chapter 9 Iterative methods

In the theory of iterative methods, two approaches to the choice of iteration parameters are in common use. The first one is related to invoking some a priori information about the operators of the iterative scheme (Bk and A in (9.2)). In the second approach (variation-type iterative methods), iteration parameters are calculated at each iteration by minimizing some functional; no a priori information about the operators is explicitly used. First, we present a general description of iteration methods without specifying the structure of the difference operators Bk . The specification of the results for the discrete elliptic operators of two-dimensional convection-diffusion problems is performed separately. As a basic problem, below, we consider the problem (9.1) with a self-adjoint positive definite operator A (A D A > 0) in a finite-dimensional Hilbert space H . We examine the iterative process B

y kC1  y k C Ay k D ', kC1

k D 0, 1, : : : ,

(9.5)

i.e., here, in contrast to the general case (9.2), the operator B is constant (independent of the iteration number).

9.1.2 Simple iteration method The simple iteration method refers to the case where the iteration parameter in (9.5) is a constant (kC1 D  ), i.e., we consider the iterative process B

y kC1  y k C Ay k D ', 

k D 0, 1, : : :

(9.6)

under the assumption that A D A > 0,

B D B  > 0.

(9.7)

The iterative method (9.6) is called a stationary method. Let a priori information about the operators B and A be given as the two-sided operator inequality (9.8) 1 B  A  2 B, 1 > 0, i.e., the operators B and A are energy equivalent with some energy equivalence constants ˛ , ˛ D 1, 2. The following basic statement is valid [131, 138]. Theorem 9.1. The iterative method .9.6/–.9.8/ converges in HD , D D A, B if 0 <  < 2= 2 . The optimal value of the iteration parameter is  D 0 D

2 , 1 C 2

(9.9)

243

Section 9.1 Basics of iterative methods

and, in the latter case, the following estimate holds for the total number n of iterations required to achieve the accuracy ": n  n0 ."/ D where 0 D

ln " , ln 0

(9.10)

1 1 , D . 1C 2

Note that, generally speaking, n0 ."/ in (9.10) is not an integer number, and n is the minimum integer number that satisfies the inequality n  n0 ."/. Theorem 9.1 shows how to optimize, according to (9.8), the iterative process (9.6), (9.7) via a proper choice of B, i.e., the operator B must be close to A in energy.

9.1.3 The Chebyshev iterative method To find the optimal set of iteration parameters in (9.5), we employ the roots of Chebyshev polynomials; that is why this method is called the Chebyshev iterative method (Richardson’s method). Define the set Mn as follows: ²   ³ 2i  1 Mn D  cos

, i D 1, 2, : : : , n . (9.11) 2n 0 , k 2 Mn , k D 1, 2, : : : , n. (9.12) 1 C 0 k The following key statement about the convergence rate of the iteration method with the Chebyshev set of iteration parameters may be formulated [131, 138]. k D

Theorem 9.2. The Chebyshev iterative method ..9.5/, .9.7/, .9.8/, .9.11/, .9.12// converges in HD , D D A, B, and the number n of iterations required to achieve the accuracy " satisfies the estimate n  n0 ."/ D where 1 D

ln.2"1 / , ln 11

(9.13)

1   1=2 1 , D . 1=2 2 1C

In the Chebyshev method, iteration parameters are evaluated (see (9.11), (9.12)) from some prescribed number of iterations n. Obviously, the degenerate case n D 1 gives the simple iteration method considered above. The practical implementation of the Chebyshev iterative method is related to the problem of computational stability. This results from the fact that the norm of the transition operator at a particular iteration may exceed unity, and therefore the local increase of the error may occur with

244

Chapter 9 Iterative methods

the following ABEND of calculations. The problem of computational stability can be resolved applying special ordering of iteration parameters, i.e., by proper choosing k from the set Mn . The optimal sequences of iteration parameters k can be calculated from the given number of iterations n by means of various algorithms (see, e.g., [138]). We note the widely-used three-level Chebyshev iterative method (see, e.g., [56, 138]), where iteration parameters are determined via recurrence relations. In this case, the error decreases monotonically, and, in contrast to using (9.11), (9.12), there is no reason to prescribe the iteration number n.

9.1.4 Two-level variation-type methods The above-considered iterative methods for solving the problem (9.1) are based on a priori information about the operators B and A given in the form of energy equivalence constants 1 and 2 (see (9.8)). Using these constants, the optimal values of iteration parameters can be determined (see (9.9), (9.12)). The derivation of these constants may be problematic, and therefore it seems reasonable to construct iterative methods where iteration parameters are determined without any a priori information. Such methods are known as variation-type iterative methods. We start with the analysis of the twolevel iterative method (9.5) under the assumption (9.7). Denote the residual and the correction by r k D Ay k  ' and w k D B 1 r k , respectively. Then the iterative process (9.5) may be written as y kC1 D y k  kC1 w k ,

k D 0, 1, : : :

It is natural to select the iteration parameter kC1 from the condition of minimal norm of the error z kC1 in HD . Direct manipulations indicate that the minimal norm is achieved with .Dw k , z k / kC1 D . (9.14) .Dw k , w k / The specification of the iterative method is achieved through the choice of D D D  > 0. This choice must be subordinated to, in particular, the possibility to calculate iteration parameters in a simple manner. The formula (9.14) involves the quantity z k that is unavailable during the computations, and therefore the simplest choice D D B (see Theorem 9.1) is impossible. The second above-mentioned possibility D D A yields the steepest descend method, where kC1 D

.w k , r k / . .Aw k , w k /

(9.15)

Among other possible choices of D, we mention the case D D AB 1 A; this method is referred to as the minimal correction method, where kC1 D

.Aw k , w k / . .B 1 Aw k , Aw k /

245

Section 9.1 Basics of iterative methods

The two-level variation-type iteration method converges no more slowly than the simple iteration method. Let us formulate this result in application to the steepest descend method. Theorem 9.3. The iterative method .9.5/, .9.7/, .9.8/, .9.15/ converges in HA , and the number n of iterations required to achieve the accuracy " satisfies the estimate .9.10/. In computational practice, the most widely-used methods are the three-level variation-type iterative methods. In the sense of the convergence rate, these methods are comparable to the iterative methods with the Chebyshev set of iteration parameters.

9.1.5 Conjugate gradient method In a three-level (two-step) iterative method, a new iteration is evaluated from the two previous iterations. To implement this technique, two initial approximations, y 0 and y 1 , are necessary. Normally, the approximation y 0 can be given arbitrarily, whereas the approximation y 1 is found using a two-level iterative method. The three-level method may be written in the following canonical form for a three-level iterative method: By kC1 D ˛kC1 .B  kC1 A/y k C .1  ˛kC1 /By k1 C ˛kC1 kC1 ', k D 1, 2, : : : ,

(9.16)

By D .B  1 A/y C 1 ', 1

0

where ˛kC1 and kC1 are iteration parameters. Calculations by the formula (9.16) are based on the representation y kC1 D ˛kC1 y k C .1  ˛kC1 /y k1  ˛kC1 kC1 w k , where, recall, w k D B 1 r k . In the conjugate gradient method, the iteration parameters for the three-level iterative method (9.16) are determined by the formulas kC1 D ˛kC1

.w k , r k / , .Aw k , w k /

k D 0, 1, : : : ,

kC1 .w k , r k / 1 D 1 k .w k1 , r k1 / ˛ k

!1

(9.17) ,

k D 1, 2 : : : ,

˛1 D 1.

The conjugate gradient method is in common use in computational practice. Theorem 9.4. Let the conditions .9.7/, .9.8/ be fulfilled. Then the conjugate gradient method .9.16/, .9.17/ converges in HA , and the number n of iterations required to achieve the accuracy " satisfies the estimate .9.13/.

246

Chapter 9 Iterative methods

We presented some reference results concerning iterative methods for solving the problem (9.1) with the self-adjoint operators A and B (the condition (9.7)). The case of non-self-adjoint problems is omitted here; the main objective of our study is the additive difference schemes.

9.2 Iterative alternating direction method This iterative method is based on the use of the ADI additive scheme. We will study the convergence rate in the case of constant (independent of the iteration number) iteration parameters. Possibilities of designing similar iterative methods are discussed for a general multicomponent splitting.

9.2.1 Iterative method with two-component splitting In a finite-dimensional Hilbert space H , we seek an approximate solution of the firstorder operator equation Ay D ' (9.18) with a positive operator A. We consider iterative methods that are based on an additive representation of the problem operator. For a two-component splitting, we have A D A1 C A2 ,

A˛ > 0,

˛ D 1, 2.

(9.19)

To specify convergence estimates, some a priori information about the operators A˛ , ˛ D 1, 2 is involved. Below, we employ the estimates A˛  ı˛ E,

ı˛ > 0,

kA˛ yk2  ˛ .A˛ y, y/,

˛ D 1, 2.

(9.20)

We study the iterative alternating direction method with a single constant iteration parameter  : y kC1=2  y k C A1 y kC1=2 C A2 y k D ',  y kC1  y kC1=2 C A1 y kC1=2 C A2 y kC1 D '. 

(9.21)

The iterative method (9.21) may be written in the canonical form for a two-level iterative method, i.e., B

y kC1  y k C Ay k D ', kC1

k D 0, 1, : : :

with the operator B D .E C A1 /.E C A2 /,

kC1 D 2 .

Section 9.2 Iterative alternating direction method

247

In the general case of the noncommutative operators A˛ , ˛ D 1, 2, the operator B is not self-adjoint and positive. This is the main obstacle to the use of the general theory of iterative methods. In fact, the same situation occurred in the study of the ADI additive difference schemes.

9.2.2 Convergence study To study the convergence rate of the iterative method (9.21), we employ the following auxiliary result. Lemma 9.1. Assume that A  ıE, ı > 0, kAyk2  .Ay, y/,

(9.22)

then the norm of the operator S. / D .E C A/1 .E  A/ is minimal under the restriction  D 0 D and kS.0 /k2 D

1 , ı

1   1=2 ı , D . 1=2

1C

(9.23) (9.24)

(9.25)

Proof. Suppose w D .E C A/y, then, by (9.23), we have S w D .E  A/y. Taking into account the representation k.E ˙ A/yk2 D kyk2 ˙ 2 .Ay, y/ C  2 kAyk2 ,

(9.26)

we get the equality k.E C A/yk2  k.E  A/yk2 D 4 .Ay, y/. Using (9.26) and the a priori information (9.22), we arrive at the estimate   1 2 2 C 2 C  .Ay, y/. k.E C A/yk  ı In view of (9.27), (9.28), we obtain k.E  A/yk2 D k.E C A/yk2  4 .Ay, y/ 1  k.E C A/yk2 , 1C

(9.27)

(9.28)

248

Chapter 9 Iterative methods

where

2 ı . 1 C ı 

D We have proved the inequality

kS wk2 

1 kwk2 , 1C

i.e.,

1 . 1C The minimum of the right-hand side occurs for  D 0 , and, in this case, the equality (9.25) holds. kS k2 

We can now return to the iterative alternating direction method. For simplicity, we restrict ourselves to the most favorable case, where ı1 1 D ı2 2 . Theorem 9.5. In solving numerically the problem .9.18/–.9.20/ with ı1 1 D ı2 2 by the iterative method .9.21/, for the number n of iterations required to ensure the accuracy " in HD  D , D D E C A2 with the iteration parameters  D 0 D

1 , ı D 1, 2 ı˛ ˛

(9.29)

the following estimate is fulfilled: n  n0 ."/ D

ln " , ln 

(9.30)

where 1=2

D

1  1 1C

!1=2

1=2 1

1=2

1  2 1C

1=2 2

!1=2 , ˛ D

ı˛ , ˛ D 1, 2.

˛

(9.31)

Proof. Consider the equation for the error of the iterative process (9.21): .E C A1 /z kC1=2 D .E  A2 /z k , .E C A2 /z kC1 D .E  A1 /z kC1=2 . Suppose w k D .E C A2 /z k , then we derive the equality w kC1 D S1 S2 w k , where

S˛ D S˛ . / D .E C A˛ /1 .E  A˛ /,

(9.32) ˛ D 1, 2.

249

Section 9.2 Iterative alternating direction method

From (9.32) follows the estimate kw kC1 k D kS1 kkS2 kkw k k. By Lemma 9.1, the norm of the operators S˛ with the iteration parameter taken in accordance with (9.29) (see (9.24)) satisfies 1=2

kS˛ k2 

1  ˛ 1C

1=2 ˛

,

˛ D

ı˛ ,

˛

˛ D 1, 2.

Therefore, each iteration gives kw kC1 k D kw k k,

(9.33)

where the constant  is defined according to (9.31). For a number of iterations, according to (9.33), we obtain the estimate (9.30). Nowadays, many variants of the iterative alternating direction method (see [138]) exist that are focused on narrower classes of problems. In particular, for the problems with self-adjoint operators A˛ , ˛ D 1, 2, extra optimization is achieved by choosing various iteration parameters in the first and the second equations of (9.21). The most preferable situation occurs in the case of the commutative self-adjoint operators A˛ , ˛ D 1, 2.

9.2.3 Modified iterative method of alternating directions The traditional iterative method of alternating directions (9.21) is associated with a pseudo-time evolution process, where, instead of the steady-state problem (9.18), we consider the Cauchy problem for the evolutionary equation dy C Ay D ', dt

t > 0.

Instead of this equation we can use a special variant of a pseudo-time evolution process, where we solve the more general equation D

dy C Ay D ', dt

t > 0,

(9.34)

with a positive operator D introduced for faster convergence of the corresponding iterative process. The choice of the operator D is not discussed here; it depends on the peculiarities of a particular problem. Consider a variant of the iterative alternating direction method based on an additive scheme for equation (9.34). This iterative method is said to be the modified method of alternating directions. For simplicity, we study the case with a self-adjoint operator D.

250

Chapter 9 Iterative methods

The iterative method under discussion has the form D

y kC1=2  y k C A1 y kC1=2 C A2 y k D ', 

y kC1  y kC1=2 C A1 y kC1=2 C A2 y kC1 D '. D 

(9.35)

Using the substitution yQ kC˛=2 D D 1=2 y kC˛=2 ,

˛ D 1, 2,

from (9.35), we obtain the system of equations yQ kC1=2  yQ k Q C AQ1 yQ kC1=2 C AQ2 yQ k D ',  yQ kC1  yQ kC1=2 Q C AQ1 yQ kC1=2 C AQ2 yQ kC1 D ',  where

AQ˛ D D 1=2 A˛ D 1=2 ,

˛ D 1, 2,

(9.36)

'Q D D 1=2 '.

The convergence of the iterative method (9.36) was studied above (see Theorem 9.5). The convergence rate is now governed by the constants ı˛ , ˛ in the inequalities AQ˛  ı˛ E,

ı˛ > 0,

kAQ˛ yk2  ˛ .AQ˛ y, y/,

˛ D 1, 2,

which may be rewritten in the equivalent form A˛  ı˛ D,

ı˛ > 0,

2 kA˛ ykD 1  ˛ .A˛ y, y/,

˛ D 1, 2.

(9.37)

Using the a priori information (9.37), we determine the optimal value of the iteration parameter and estimate the iterations number for the modified iterative method of alternating directions (9.35).

9.2.4 Multicomponent splitting The optimization of iterative methods (minimization of the computational costs) is achieved by a proper choice of iteration parameters and preconditioners. Currently, much attention is paid to the choice of preconditioners in the form of the product of two economical operators. In fact, we speak of using some variants of the classical two-component iterative method of alternating directions. For three-dimensional problems, the use of domain decomposition methods and multicolor iterative methods is often focused on multicomponent splitting rather than

251

Section 9.2 Iterative alternating direction method

on two-component decomposition. In [149], at the general operator level, possibilities are shown for constructing two basic variants of iterative methods with multicomponent splitting. In the first variant (iterative methods with a multiplicative preconditioner), a new iteration is obtained using the sequential solution of elementary problems. The second technique (additive algorithms) allows parallel computing. The fundamental point is connected with the fact that in the class of iterative methods under consideration, for solving problems with a self-adjoint positive operator, we can use three-level iterative variation-type methods, i.e., the standard acceleration via the conjugate gradients can be applied. For instance, assume that we seek the solution of the system of linear equations (9.18) with a positive and self-adjoint operator A. To construct an iterative method, we involve the representation AD

p X

A˛ > 0,

A˛ D A˛  0,

˛ D 1, 2, : : : , p.

(9.38)

˛D1

The traditional iterative method of alternating directions is connected with the choice of the operator B that has the following multiplicative representation: BD

p Y

.E C !˛ A˛ /,

(9.39)

˛D1

with some positive constants !˛ , ˛ D 1, 2, : : : , p. The optimization of numerical techniques by means of a proper choice of the iteration parameters in (9.39) is achieved only under the assumption of the pairwise commutativity of the operators A˛ , ˛ D 1, 2, : : : , p. In the general case of noncommutative operators, even for twocomponent splitting, we have B ¤ B  . We start with the construction of the iterative methods of multicomponent splitting (9.38) with multiplicative preconditioners (the class of the factorized operators B). Consider the iterative method p Y

.E C !˛ A˛ /

˛D1

p Y y kC1  y k C .E C !˛ A˛ /1 .Ay k  '/ D 0. kC1 ˛D1

(9.40)

The equation (9.40) may be written in the canonical form BD

p Y ˛D1

.E C !pC1˛ ApC1˛ /

p Y

.E C !˛ A˛ /,

(9.41)

˛D1

and therefore B D B  > 0. In comparison with the traditional iterative method of multicomponent splitting (9.38), (9.39), we have a doubling of the computational costs for one iteration. But at the same time, the symmetrized iterative method (9.38), (9.41) makes it possible to optimize the choice of iteration parameters for the general case of noncommutative operators A˛ , ˛ D 1, 2, : : : , p.

252

Chapter 9 Iterative methods

If we focus on modern parallel computers, our main attention is on iterative methods that have additive rather then multiplicative representation of the operator B 1 . Instead of (9.40), we use the iterative method X y kC1  y k C .E C !˛ A˛ /1 .Ay k  '/ D 0. kC1 ˛D1 p

(9.42)

In the canonical form for the iterative method (9.42), we have B 1 D

p X

.E C !˛ A˛ /1

(9.43)

˛D1

and then again B D B  > 0. The implementation of the iterative method (9.42) can be conducted as .E C !˛ A˛ / y kC1 D

y˛kC1  y k C Ay k D ', kC1

p 1 X kC1 y . p ˛D1 ˛

Such a scheme is typical for additively averaged schemes. The iterative method can be treated as a vector additive scheme. Here we restrict ourselves to the general description of possible iterative schemes of multicomponent splitting. Under the fairly general assumptions (9.38), we noted the possibility of using serial (multiplicative) and parallel (additive) computational algorithms with a self-adjoint and positive definite preconditioner. For more specific classes of splitting, it is necessary to formulate and to solve the problem of optimization of iterative methods; we need to investigate the convergence rate.

9.3 Iterative alternating triangle method For the operator B, we can take the operator that corresponds to the lower (or upper) triangular matrix; moreover, it may be the product of operators with such a structure. Iterative methods from this class of numerical techniques are very popular in applied calculations.

9.3.1 Iterative method In a finite-dimensional Hilbert space H , we seek the solution of the equation Ay D ',

(9.44)

253

Section 9.3 Iterative alternating triangle method

where the operator A D A > 0. In the iterative alternating triangle method, the following two-component additive splitting is used: A D A1 C A2 > 0,

A1 D A2 .

(9.45)

To solve numerically the problem (9.44), (9.45), we apply the two-level iterative method y kC1  y k B C Ay k D ', k D 0, 1, : : : . (9.46) kC1 In the iterative alternating triangle method [118], the operator B, in accordance with the operator splitting (9.45), has the form B D .E C !A1 /.E C !A2 /

(9.47)

with some parameter ! > 0. For the additive scheme of alternating triangles, we have kC1 D  , ! D 0.5 . In solving the system of linear algebraic equations (9.44) (the operator A is a matrix), the operator B, defined according to (9.47), is the product of two triangular matrices. For the splitting (9.45), the operator B is self-adjoint, and so we can use general results of the theory of iterative methods. It is possible to increase the convergence rate of iterative methods by a proper choice of the iteration parameters kC1 using, e.g., the conjugate gradient method.

9.3.2 Convergence rate The convergence rate of the iterative method (9.46), (9.47) is governed by the constants of the energy equivalence ˛ , ˛ D 1, 2 in the bilateral operator inequality 1 B  A  2 B,

1 > 0.

(9.48)

We find these constants using a priori information in the form of the inequalities A  ıE,

A1 A2 

A. 4

(9.49)

For the operator B, taking into account the representation (9.45), (9.47), we have B D E C !A C ! 2 A1 A2 .

(9.50)



 1 2 B C ! C ! A. ı Thus, the constant 1 is determined as

In view of (9.49), we get

1 D

ı . 1 C !ı C ! 2 ı =4

(9.51)

254

Chapter 9 Iterative methods

To evaluate the constant 2 , we represent the operator B using (9.50) as follows: B D E  !.A1 C A2 / C ! 2 A1 A2 C 2!A D .E  !A1 /.E  !A2 / C 2!A. Hence, we have .By, y/  2!.Ay, y/, i.e., A  2 B, where 1 . (9.52) 2 D 2! Now we can select the parameter ! in (9.47) from the condition of the maximum of  D .!/ D 1 = 2 . In view of (9.51), 9.52), we obtain .!/ D

2!ı 1 D . 2 1 C !ı C ! 2 ı =4

The maximal .!/ occurs with ! D !0 D

2 , .ı /1=2

(9.53)

and

21=2 ı , D . (9.54) 1=2

1C Based on the obtained estimates, the following statement about the convergence of the iterative alternating triangle method with the optimal value of ! may be formulated.  D .!0 / D

Theorem 9.6. The iterative alternating triangle method .9.45/–.9.47/, .9.53/ with the Chebyshev set of iteration parameters converges in HA and HB , and, for the total number of iterations required to ensure the accuracy ", the following estimate is valid: n  n0 ."/ D where 1 D

ln.2"1 / , ln 11

(9.55)

1   1=2 1 , D , 1=2 2 1C

and  is determined by .9.54/. For small , the number of iterations can be reduced to the simpler expression ln.2"1 / ı n0 ."/ D p 1=4 ,  D .

2 2 The iterative alternating triangle method can be implemented using the conjugate gradient method. In this case, the number of iterations is characterized by the above estimate (9.55). In numerically solving boundary value problems for elliptic equations of second order, the iteration number for the iterative alternating triangle method is proportional to the square root of the number of grid points in a single spatial direction [138].

Section 9.4 Iterative cluster aggregation methods

255

9.3.3 Modified iterative method of alternating triangles Let us discuss a possible modification of the iterative alternating triangle method, which may be attributed to calculations in order to establish the steady-state solution for the equation dy D C Ay D ', t > 0. dt We choose the operator B in the following factorized form: B D .D C !A1 /D 1 .D C !A2 /,

(9.56)

where D D D  > 0. The most important properties – self-adjointness and positiveness – are preserved for the operator B in the generalization (9.56). The iteration number for the modified iterative method of alternating triangles (9.45), (9.46), (9.56) satisfies the estimate that is similar to the restriction provided by Theorem 9.6. The convergence rate depends on the parameters ı, that are involved into the inequality

A  ıD, A1 D 1 A2  A. 4 The proof of this fact is conducted similarly to Theorem 9.6.

9.4 Iterative cluster aggregation methods The theory of iterative methods for solving systems of linear equations is developing in various directions [56, 114, 138]. If we focus on modern parallel computers [27, 103], then success is achieved through the use of the classical block iterative methods, techniques with multicolor ordering of unknowns. In solving elliptic boundary value problems, approaches based on decomposition (splitting) of a problem domain into overlapping or non-overlapping subdomains are considered. The alternating Schwarz method provides the classical example of domain decomposition techniques. At the matrix level, domain decomposition methods may be treated as special iterative algorithms of block type. Following the work [144], we distinguish a class of iterative techniques with special algorithms that are typical for the classical block methods. Individual equations of the system are aggregated into groups after some pre-processing procedure, e.g., after scaling. Various groups, which are called clusters, may include common equations. For a symmetric linear system of equations, convergence of iterative cluster aggregation methods has been proved. Among them we highlight the methods that are associated with the methods of point or block relaxation, the iterative domain decomposition methods of Schwarz type.

256

Chapter 9 Iterative methods

9.4.1 Transition to a system of equations In a finite-dimensional Hilbert space H , we seek the solution of the equation Ay D '

(9.57)

with a positive and self-adjoint operator A. So far, we have considered iterative methods that are associated with the additive representation of the problem operator in the form p X  A˛ > 0, A˛  0, ˛ D 1, 2, : : : , p. (9.58) ADA D ˛D1

Such a representation results from, e.g., splitting into one-dimensional operators – splitting with respect to the spatial variables. The second class of methods under discussion is connected with the splitting of the problem operator into two operators that are adjoint to each other – we speak of the alternating triangle decomposition. Here we present a new class of splittings that allows us to include different variants of block iterative methods into a general scheme for constructing iterative methods on the basis of an additive representation of a problem operator. We confine ourselves to general considerations at the operator level omitting details of particular implementations. Assume that (9.57) is a system of linear equations, then the class of iterative methods studied here is based on combining (aggregation) of separate equations of the system into clusters. It should be noted that different clusters that appeared after some transformations may include common equations. The classical block methods of linear algebra give us examples of this approach. Suppose that from the single equation (9.57), we obtain a system of p equations (p clusters). The construction of the system of equations may be formalized through the introduction of the aggregation operators G˛ , ˛ D 1, 2, : : : , p. Under sufficiently general conditions, we assume that G˛ D G˛  0,

˛ D 1, 2, : : : , p,

GN D

p X

G˛ > 0.

(9.59)

˛D1

To get separate equations, we multiply the original equation (9.57) by G˛ , ˛ D 1, 2, : : : , p which yields G˛ Ay D G˛ ',

˛ D 1, 2, : : : , p.

Thus, we pass from equation (9.57) to the system of equations A ˛ y D '˛ ,

˛ D 1, 2, : : : , p,

(9.60)

where A˛ D G˛ A,

'˛ D G˛ ',

˛ D 1, 2, : : : , p.

(9.61)

257

Section 9.4 Iterative cluster aggregation methods

Let y be a vector that satisfies (9.60), (9.61). Then the summation over all ˛ D 1, 2, : : : , p yields N N GAy D G'. Under our assumptions, GN > 0, and therefore each solution of (9.60), (9.61) is just the unique desired solution of the original system of equations (9.57).

9.4.2 Iterative method To solve numerically the system of equations (9.57), we use an iterative method that appears due to cluster aggregation during the transition to the system of equations in accordance with (9.57)–(9.61). That is why the methods of this class are referred to as iterative cluster aggregation methods. The transition from the iteration y k to the next iteration y kC1 involves the solution of p problems that meet the cluster decomposition (9.60), (9.61). The approximate solution that corresponds to the cluster (equation) with the index ˛ is denoted by y kC˛=p . We start with the method where y kC˛=p is determined sequentially with increasing the index ˛. In this case, we speak of a synchronous iterative method. The approximate solution y kC˛=p is evaluated from the previously found iteration kC.˛1/=p . For a stationary iterative method, we put y .E C A˛ /

y kC˛=p  y kC.˛1/=p C A˛ y kC.˛1/=p D '˛ , 

(9.62)

˛ D 1, 2, : : : , p. Here  is a positive constant (a parameter of the iterative method). The iterative method (9.62) may be associated with the use of an additive scheme of componentwise splitting for solving the Cauchy problem for the equation 

dy C Ay D ', dt

t > 0.

Theorem 9.7. The iterative cluster aggregation method .9.58/–.9.62/ converges in HA for any 0 <  < 2. Proof. Consider the corresponding problem for the error z kC˛=p D y kC˛=p  y, ˛ D 1, 2, : : : , p. In view of (9.61), we have .E C G˛ A/

z kC˛=p  z kC.˛1/=p C G˛ Az kC.˛1/=p D 0, 

(9.63)

˛ D 1, 2, : : : , p. Multiplying (9.63) by A, rewrite this system in the form     1 1 z˛ C 1  z˛1 D 0, AztN,˛ C AG˛ A  

(9.64)

258

Chapter 9 Iterative methods

where z˛ D z kC˛=p ,

z˛  z˛1 , 

ztN,˛ D

˛ D 1, 2, : : : , p.

Scalar multiplication of (9.64) by 2 z˛ D 2z˛ C .2  2/z˛1 D  .z˛ C z˛1 /  .  2/ ztN,˛ , yields the equality 2 2 2   kz˛1 kA C .2   /  kztN,˛ kA C 2 .˛ Az˛ , Az˛ / D 0,  kz˛ kA

˛ D 1, 2, : : : , p.

(9.65)

Adding all equations (9.65), we obtain p X  2 2 2  zp A   kz0 kA C .2   /   0. kztN,˛ kA

(9.66)

˛D1

It is easy to see that p X

2 > 0. kztN,˛ kA

(9.67)

˛D1

In the opposite case, i.e., if the sum is equal to zero, we have z kC˛=p D z k ,

˛ D 1, 2, : : : , p.

G˛ Az k D 0,

˛ D 1, 2, : : : , p.

By (9.62), we get Under the restrictions (9.59), the recent equality is true if and only if z k D 0, i.e., it is valid for the exact solution of the original problem only. From (9.66) and (9.67), with 0 <  < 2, we get in the original notation:        kC1  (9.68)  < z k  . z A

A

This estimate ensures the convergence of the iterative process (9.62) to the solution of the problem (9.57) in HA . The most important issue is to study the convergence rate. An investigation of its dependence on the operators G˛ , ˛ D 1, 2, : : : , p and the numerical parameter  requires a deep analysis for each particular variant of aggregation. A similar remark can be also applied to the problem of optimal choice of the iteration parameter  or variable iteration parameters kC˛=p , ˛ D 1, 2, : : : , p; this problem is beyond the present discussion.

259

Section 9.4 Iterative cluster aggregation methods

9.4.3 Parallel variant We focus on constructing computational algorithms for up-to-date parallel computers, and therefore special attention should be given to asynchronous iterative methods. In this case, the approximate solution is calculated on the basis of individual subproblems that can be solved independently of each other. Let us discuss some opportunities in this area of research. Define the vectors y˛kC1 from the equations .E C A˛ /

y˛kC1  y k C A ˛ y k D '˛ , 

˛ D 1, 2, : : : , p.

(9.69)

For the approximate solution at the k C 1-st iteration, we use the expression y kC1 D

p 1 X kC1 y . p ˛D1 ˛

(9.70)

The principal discrepancy between the algorithm (9.69), (9.70) and the method (9.62) is that the determination of y˛kC1 can be performed independently (asynchronously) of each other using the iterative approximation y k only. The convergence of this variant is established under the same conditions as for the synchronous version of the above-discussed iterative method of cluster aggregation. Theorem 9.8. The iterative cluster aggregation method .9.58/–.9.61/, .9.69/, .9.70/ converges in HA under the condition 0 <  < 2. Proof. Similarly to the proof of Theorem 9.7, under the above restrictions on  , we have  2  2     2 C 2  0,  z˛kC1    z k  C .2   /  kQztN,˛ kA A

A

where now  z˛ D z˛kC1 C .  1/z k , z˛kC1  z k .  Note also that, at least for one of ˛ D 1, 2, : : : , p, the last inequality is strict. In view of (9.70), we have z˛kC1 D y˛kC1  y,

zQ tN,˛ D

p    1 X  kC1 2  kC1 2   y˛  , y A A p ˛D1

and the desired estimate (9.68) (9.68) holds.

260

Chapter 9 Iterative methods

Synchronous versions of the cluster aggregation method may be treated as analogs of iterative procedures of Gauss–Seidel type, whereas the asynchronous variant is naturally associated with iterative methods of Jacobi type. The possibility to arrange parallel computing results, generally speaking, in a slower rate of convergence. As for correlations between iterative methods and additive schemes, we note that the iterative method (9.69), (9.70) is a prototype of the additively averaged scheme of multicomponent splitting.

9.4.4 Aggregation of unknowns Let us discuss the possibility of constructing iterative cluster aggregation methods from a slightly different point of view. We have already considered some variants for the transition from the original equation to a system of equations, i.e., for aggregation of equations. The second approach (see [192]) is associated with the separation of unknowns – aggregation of unknowns. Represent the solution of equation (9.57) in the form p X G˛ y˛ . (9.71) yD ˛D1

The substitution into (9.57) yields p X

AG˛ y˛ D f .

(9.72)

˛D1

Instead of the single unknown y, in equation (9.57), now we have in (9.72) p unknowns y˛ , ˛ D 1, 2, : : : , p. The function y that is determined in accordance with (9.72) may be treated as the solution of the original problem (9.57). It is clear that the new problem (9.72) has many solutions. To find any of them, we use an iterative method. The approximate solution y˛ at the k-th iteration is denoted by y˛k . The next iteration is determined from the system of equations p X y˛kC1  y˛k AGˇ yˇk D f . C .E C AG˛ / 

(9.73)

ˇ D1

This iterative process with cluster aggregation of unknowns is very similar in organization to the iterative method with cluster aggregation of equations in the form (9.69), (9.70). Theorem 9.9. The iterative cluster aggregation method .9.59/, .9.73/ converges in HA for any 0 <  < 2=p.

261

Section 9.4 Iterative cluster aggregation methods

Proof. We outline the main points for the proof of this statement. Considering the problem for the error, from (9.73), we obtain .E C AQ˛ /

X v˛kC1  v˛k AQˇ vˇk D 0, C  p

(9.74)

ˇ D1

where

v˛k D A1=2 .y˛k  y˛ /, AQ˛ D A1=2 G˛ A1=2 . Multiplying equation (9.74) by AQ˛ , we rewrite it in the form .AQ˛ C AQ2˛ /

p p X X v˛kC1  v˛k 1 1 C AQ˛ AQˇ .vˇkC1 C vˇk /  AQ˛ AQˇ .vˇkC1  vˇk / D 0.  2 2 ˇ D1

ˇ D1

If we multiply scalarly each equation by 2.v˛kC1  v˛k / D 2 v t,˛ and summarize them, then we get 2

p X

.AQ˛ v t,˛ , v t,˛ / C 2

˛D1

C

X p ˛D1

AQ˛ v˛kC1 ,

p X

.AQ˛ v t,˛ , AQ˛ v t,˛ /  

˛D1 p X



AQ˛ v˛kC1 

X p

AQ˛ v t,˛ ,

p X

˛D1 ˛D1  p X AQ˛ v˛k , AQ˛ v˛k D ˛D1 ˛D1

X p

˛D1

AQ˛ v t,˛



0. (9.75)

Similarly to (9.71), if we introduce yk D

p X

G˛ y˛k ,

˛D1

then, for the last terms in (9.75), we obtain p X

AQ˛ v˛k D A1=2 .y k  y/.

˛D1

Taking into account the inequality X 2 p p X Q .AQ˛ v t,˛ /2 , A˛ v t,˛  p ˛D1

˛D1

from (9.75) follows the inequality 2

p X ˛D1

.AQ˛ v t,˛ , v t,˛ / C .2  p /

p X

.AQ˛ v t,˛ , AQ˛ v t,˛ /

˛D1

C .A.y kC1  y/, .y kC1  y//  .A.y k  y/, .y k  y//  0. This estimate, similarly to the proof of Theorem 9.9, proves the convergence of the iterative process (9.73).

262

Chapter 9 Iterative methods

The iterative cluster aggregation process is closely related to vector additive schemes of multicomponent splitting for numerically solving the Cauchy problem for the equation p X dy A˛ y D ', t > 0,  C dt ˛D1 where, by (9.72), we put A˛ D AG˛ ,

˛ D 1, 2, : : : , p.

In this regard, we highlight the possibility of using iterative methods that are based on sequential computations p ˛1 X X y˛kC1  y˛k k AGˇ yˇ C AGˇ yˇk D f . C .E C AG˛ /  ˇ D1

ˇ D˛

It should be noted that the methods of this class can be accelerated using the conjugate gradient method.

Chapter 10

Additive schemes with splitting of the operator at the time derivative Here additive schemes are constructed using a splitting of a problem operator into the sum of operators. The transition to a new time level is conducted by means of solving simpler problems that are associated with individual operator terms of the problem operator. We consider a new class of additive schemes for an operator-differential evolutionary equation of first order with an additive representation of the operator at the time derivative. Vector operator-difference schemes, which are characterized by a transition from the single original evolutionary equation to a system of evolutionary equations, are designed and investigated. On the basis of additive schemes (operator-splitting schemes), we create efficient computational algorithms to solve initial-boundary value problems for systems of time-dependent PDEs. We study problems in which the components of the solution vector are interconnected with respect to the time derivatives. Splitting schemes based on an additive representation for both the leading problem operator and the operator at the time derivative are proposed; they employ a triangular two-component representation of the operators.

10.1 Schemes with splitting of the operator at the time derivative Nowadays, various classes of additive operator-difference schemes for evolutionary equations are constructed using additive splitting into several terms for the leading operator, which acts on the solution itself rather than on its time derivative. For a number of applications, it is interesting to investigate the problems that have an additive representation for the operator at the time derivative. Below, for this new class of evolutionary problems, we will develop and analyze vector operator-difference schemes.

10.1.1 Preliminary discussions To solve numerically multidimensional transient problems of mathematical physics, different classes of additive schemes (operator-splitting schemes) [93, 131, 218] are widely used. Beginning with the pioneer works [28, 104], the simplest way to construct additive schemes is in the splitting of the problem operator into the sum of two operators with a simpler structure – we speak of ADI methods, factorized schemes, predictor-corrector techniques and so on [151].

264

Chapter 10 Splitting of the operator at the time derivative

In the more general case of multicomponent splitting, the existing classes of unconditionally stable operator-difference schemes are based on the concept of summarized approximation. In this way, we can construct the classical locally one-dimensional schemes (schemes of componentwise splitting) [93, 131], additively averaged locally one-dimensional schemes [50, 151] etc. A new class of unconditionally stable schemes – vector additive schemes (multicomponent schemes of alternating directions) – is now actively developed (see, e.g., [1, 193]). These schemes have full approximation, i.e., each intermediate problem approximates the original one. The simplest way to construct additive schemes of full approximation is based on the regularization principle for operator-difference schemes. Improving quality of operator-difference schemes is achieved applying additive or multiplicative perturbations of operators of the scheme [126]. Regularized additive schemes for evolutionary equations of first and second order are developed for equations as well as for systems of equations [158, 202]. First, the standard schemes of splitting with respect to separate directions (locally one-dimensional schemes) and splitting with respect to physical processes are considered [131, 151]. Secondly, regionally additive schemes based on domain decomposition are studied for constructing parallel algorithms for transient problems of mathematical physics [96, 136, 196]. The above-mentioned classes of additive operator-difference schemes for evolutionary equations are based on an additive splitting of the leading operator into several terms. For many problems of practical interest, it is interesting to investigate the problems that have an additive representation for the operator at the time derivative. In the first publication on this subject [205], vector additive operator-difference schemes were proposed and examined in which the operator at the time derivative was split into the sum of self-adjoint and positive definite operators. Following this paper, we investigate vector additive schemes for such problems.

10.1.2 Statement of the problem Let H be a finite-dimensional Hilbert space, and A, B, D be linear operators in H . We consider grid functions y from the finite-dimensional real Hilbert space H , where the scalar product and norm are denoted by ., / and kyk D .y, y/1=2 , respectively. For D D D  > 0, we introduce the space HD equipped with scalar product .y, w/D D .Dy, w/ and the norm kykD D .Dy, y/1=2 . In the Cauchy problem for an evolutionary equation of first order, we seek the function y.t / 2 H that satisfies du C Au D f .t /, dt with a given f .t / 2 H and the initial data B

u.0/ D u0 .

t >0

(10.1)

(10.2)

Section 10.1 Schemes with splitting of the operator at the time derivative

265

Such a problem occurs in many applications. First of all, we speak of mathematical models that are based on boundary value problems for equations of Sobolev type. In the work [167], the following equation:   2 @2 u @2 u @2 @2 u 2@ u C C D 0, C ˛ @t 2 @x12 @x22 @x32 @x32 was obtained for describing oscillations in a rotating fluid. Models with the time derivative of the first order have often been studied for the pseudo-parabolic equation du C Lu D 0, dt where M and L are elliptic operators [163, 164]. The most comprehensive review of applied problems connected with equations of Sobolev type is presented in the recently published book [177]. Beginning with the pioneer works [32,33], various numerical algorithms have been developed for equations of Sobolev type. At present, different classes of additive operator-difference schemes for evolutionary equations are obtained via an additive decomposition of the operator L into several terms; this operator is associated with the solution u rather than with its time derivative. For a number of applications, it is more interesting to consider problems in which an additive representation does exist for the operator at the time derivative, i.e., for the operator M . We have such a situation, e.g., for the above-mentioned Sobolev equation, where the operator at the time derivative is the sum of three one-dimensional operators and the operator at u is one-dimensional, too. Assume that the linear operators A and B, acting from H onto H (A : H ! H , B : H ! H ), are positive, self-adjoint and stationary; i.e., d d d d A D A , B D B  > 0, BDB . A D A > 0, dt dt dt dt For the problem (10.1), (10.2), we can obtain various a priori estimates which express stability of the solution with respect to the initial data and the right-hand side in different spaces. We restrict ourselves to elementary estimates, trying to get the same type of estimates for both scalar and vector problems as well as for the solution of both differential and discrete problems. Multiplying scalarly both sides of equation (10.1) in H by u, we get 1d .Bu, u/ C .Au, u/ D .f , u/. 2 dt For the right-hand side, we use the estimate  1  1 A f ,f . .f , u/  .Au, u/ C 4 This yields the following a priori estimate for the solution of the problem (10.1), (10.2): Z 1 t 2 0 2 2 ku.t /kB  ku kB C kf .s/kA (10.3) 1 ds, 2 0 M

266

Chapter 10 Splitting of the operator at the time derivative

which expresses stability with respect to the initial data and the right-hand side. Standard additive schemes are characterized by using the splitting of the operator A into the sum of operators with a simpler structure. For instance, assume that the operator A satisfies the following additive representation: AD

p X

A˛ ,

A˛ D A˛  0,

˛ D 1, 2, : : : , p.

(10.4)

˛D1

Additive difference schemes are constructed on the basis of (10.4), where the original problem is decomposed into p subproblems. The transition from the time level tn to the next level t nC1 D t n C  , where  > 0 is a time step and y n D y.t n /, t n D n , n D 0, 1, : : :, is associated with solving problems for the individual operators A˛ , ˛ D 1, 2, : : : , p in the additive decomposition (10.4). The subject that we consider will be another case. In a number of problems, computational complexity is attributed to the operator B staying at the derivative in time, rather than with the operator A acting on the solution. In this case, to decrease the computational complexity of the problem (10.1), (10.2), we employ the additive representation p X B˛ , B˛ D B˛ > 0, ˛ D 1, 2, : : : , p (10.5) BD ˛D1

instead of (10.4). The transition to a new time level is connected with the solution of some auxiliary Cauchy problems for the equations du˛ C Au˛ D f˛ .t /, dt with proper specified initial data. B˛

t > 0 ˛ D 1, 2, : : : , p

10.1.3 Vector problem By definition, put u D ¹u1 , u2 , : : : , up º. Each individual component is defined as the solution of the similar problems p X ˇ D1

Bˇ

duˇ C Au˛ D f .t /, dt

u˛ .0/ D u0 ,

t > 0,

˛ D 1, 2, : : : , p.

(10.6) (10.7)

Here is an elementary coordinatewise estimate for stability of the solution. Subtracting one equation from another, we get A.u˛  u˛1 / D 0,

˛ D 2, 3, : : : , p.

Taking into account the positiveness of the operator A, this gives u˛ D u˛1 ,

˛ D 2, 3, : : : , p.

Section 10.1 Schemes with splitting of the operator at the time derivative

267

For the individual component u˛ , we obtain the same equation as for u, i.e. p X

Bˇ

ˇ D1

du˛ C Au˛ D f .t /, dt

t > 0,

˛ D 1, 2, : : : , p.

For the same reason, the a priori estimates are satisfied Z 1 t 2 0 2 2 kf .s/kA ˛ D 1, 2, : : : , p. ku˛ .t /kB  ku kB C 1 ds, 2 0

(10.8)

It follows that u˛ .t / D u.t /,

t > 0,

˛ D 1, 2, : : : , p.

Therefore, as the solution of the original problem (10.1), (10.2), we can take any component of the vector u.t /. For the vector evolutionary problem, we can derive (see, e.g., [151] for the additive schemes with the splitting (10.4)) a priori estimates for the vector u considering the problem in the Hilbert space H D H p with the scalar product .u, v/ D

p X

.u˛ , v˛ /.

˛D1

We rewrite equation (10.6) in the form B˛ A

1

p X ˇ D1

Bˇ

duˇ C B˛ u˛ D fQ˛ .t /, dt

t > 0,

˛ D 1, 2, : : : , p,

where fQ˛ D B˛ A1 f . This allows us to write the system of equations in the vector form du C C Du D fQ . (10.9) dt The operator matrices C and D have the form C D ¹C˛ˇ º,

C˛ˇ D B˛ A1 Bˇ ,

D D ¹D˛ˇ º,

D˛ˇ D B˛ ı˛ˇ ,

(10.10) ˛, ˇ D 1, 2, : : : , p,

where ı˛ˇ is Kronecker’s symbol. The equation (10.9) is supplemented with the initial data u.0/ D u0 . (10.11) The principal advantage of the formulation (10.9) results from the fact that C D C   0, in H .

D D D > 0

268

Chapter 10 Splitting of the operator at the time derivative

Here is an a priori estimate for the solution of the vector problem (10.9)–(10.11). This estimate, on the one hand, is more complicated than (10.8) and, on the other hand, we will use it as a guide in the corresponding operator-difference schemes. Multiplying both sides of (10.9) scalarly in H by d u=dt , we get     du 1d du du , C .Du, u/ D fQ , . (10.12) C dt dt 2 dt dt In view of (10.10), we obtain     p p X X du du 1 Bˇ u ˇ , Bˇ u ˇ , , D A C dt dt ˇ D1

ˇ D1

and, for the right-hand side of (10.12), we have       p X  1  1 du du du 1 Q Bˇ u ˇ  C D A f, , C A f ,f . f, dt dt dt 4

(10.13)

ˇ D1

Similarly to (10.3), (10.8), from (10.12), (10.13), it follows that the estimate Z 1 t 2 kf .s/kA (10.14) kuk2D  ku0 k2D C 1 ds 2 0 holds. Taking into account (10.10), we get kuk2D D

p X

.B˛ u˛ , u˛ / .

˛D1

Thus, the estimate (10.14) may be treated along with (10.8) as a vector analog of the estimate (10.3). By (10.5), the estimate (10.12) ensures the stability of any individual component of the vector u.t /.

10.1.4 Vector additive schemes Splitting schemes for numerically solving the problem (10.1), (10.2), (10.5) will be constructed using usual schemes with weights for the vector problem (10.6), (10.7). The standard two-level scheme with weights for the problem (10.1), (10.2) has the form y nC1  y n C A.y nC1 C .1   /y n / D ' n , n D 0, 1, : : : ,  where, e.g., ' n D f . t nC1 C .1   /t n /, B

(10.15)

and  is a weight parameter (usually 0    1). In the general stability theory of operator-difference schemes developed by Samarskii [131, 134, 136], the exact (not improvable) stability criteria for two- and threelevel operator-difference schemes were obtained in various norms. They can be used directly in the investigation of schemes with weights (10.15). Here is a typical result.

269

Section 10.1 Schemes with splitting of the operator at the time derivative

Theorem 10.1. If   1=2, then the operator-difference scheme .10.15/ is absolutely stable in HB , and the difference solution satisfies the levelwise estimate  2 2 2 ky nC1 kB  ky n kB C k' n kA 1 . 2

(10.16)

Proof. By definition, put  1 1  y nC1  y n y .n/ D y nC1 C .1   /y n D .y nC1 C y n / C    . 2 2  Multiplying both sides of (10.15) by y .n/ scalarly in H , we get 1 .B.y nC1  y n /, y nC1 C y n / 2    1 y nC1  y n y nC1  y n C   , B 2   C .Ay .n/ , y .n/ / D .' n , y .n/ /. For the right-hand side, we apply the estimate 1 .' n , y .n/ /  .Ay .n/ , y .n/ / C .A1 ' n , ' n /. 4 If   1=2, then we obtain the desired estimate (10.16) for stability of the approximate solution with respect to the initial data and the right-hand side, which is a discrete analog of the estimate (10.3) for the solution of the problem (10.1), (10.2). This concludes the proof. To solve the vector problem (10.6), (10.7), we apply the following difference scheme:   nC1 p X yˇn  yˇn1 y˛n  y˛n1 y˛  y˛n Bˇ C .1   / C B˛     ˛¤ˇ D1 (10.17) C A.y˛nC1 C .1  2 /y˛n C y˛n1 / D ' n , n D 0, 1, : : : ,

˛ D 1, 2, : : : , p.

Unlike (10.15), the scheme (10.17) is a three-level one and has two weight factors:  and  . The numerical implementation of the scheme (10.17) involves sequentially solving discrete problems   n B˛ C  A y˛nC1 D n˛ , ˛ D 1, 2, : : : , p during the transition from the time level t n to the new level t nC1 . For the vector additive scheme (10.17), we can arrange parallel computing, i.e., independent calculations of the individual components.

270

Chapter 10 Splitting of the operator at the time derivative

Using notation (10.10), we write the operator-difference scheme (10.17) in the vector form G

y n  y n1 y nC1  2y n C y n1 CC   C D. y nC1 C .1  2 /y n C  y n1 / D g n ,

(10.18)

where G D ¹G˛ˇ º, g n D ¹g˛n º,

G˛ˇ D B˛ A1 B˛ ı˛ˇ , g˛n D B˛ A1 ' n ,

˛, ˇ D 1, 2, : : : , p.

Thus, in (10.18), we have that the operator G D G  > 0. Taking into account that y nC1  y n1 y nC1  2y nC1 C y n1 y n  y n1 D  ,  2 2  y nC1 C .1  2 /y n C  y n1 D   1 1 .y nC1  2y nC1 C y n1 / C .y nC1 C 2y nC1 C y n1 /,  4 4 rewrite (10.18) in the form C

y nC1  2y nC1 C y n1 y nC1  y n1 CR 2 

(10.19)

1 C D.y nC1 C 2y nC1 C y n1 / D g n , 4  1 1 D. R D G  C C    2 4

where Let

1 vn D .y n C y n1 /, 2

wn D y n  y n1

and write (10.19) as wnC1 C wn wnC1  wn 1 CR C D.vnC1 C y n / D g n . 2  2 Multiplying both sides of (10.20) scalarly by C

(10.20)

2.vnC1  vn / D wnC1 C wn , we get the equality 1 1 .C .wnC1 C wn /, wnC1 C wn / C .R.wnC1  wn /, wnC1 C wn / 2  C .D.vnC1 C vn /, vnC1  vn / D .g n , wnC1 C wn /.

(10.21)

Section 10.1 Schemes with splitting of the operator at the time derivative

271

Similarly to (10.13), we have 1  .C .wnC1 C wn / C .A1 ' n , ' n /. 2 2 With this in mind, from (10.21), it follows that  (10.22) E nC1  E n C .A1 ' n , ' n /, 2 where 1 E n D .Dvn , vn / C .Rwn , wn /.  We formulate the conditions under which the value of E n determines the square of the norm of the difference solution. By virtue of the positiveness of the operator D, it is sufficient to require non-negativity of the operator R. For the energy of the operators C and G , the following coordinatewise representation holds:   p p X X 1 B˛ u ˛ , B˛ u ˛ , .C u, u/ D A .g n , wnC1 C wn / 

˛D1

.Gu, u/ D

p X 

˛D1

 A1 B˛ u˛ , B˛ u˛ .

˛D1

Considering

   X 2  p p p X X A1 B˛ u ˛ , B˛ u˛ D A1 B˛ u ˛ , 1 ˛D1

˛D1

˛D1

p Dp

p X



˛D1 p X



 A1 .B˛ u˛ /2 , 1  A1 B˛ u˛ , B˛ u˛ ,

˛D1

we have C  pG . Therefore, for   1=4 and   p=2, it follows that R  0. Thus, we have proved the following statement. Theorem 10.2. If   1=4 and   p=2, then the operator R  0 in H , and the vector additive scheme .10.17/ is absolutely stable such that the difference solution satisfies the a priori estimate .10.22/ with   n  y C y n1 2   n  C 1 R.y n  y n1 /, y n  y n1 .  E D  2  D The above-proved a priori estimate (10.22) guarantees the stability of the difference solution at the half-integer time levels (for vn ) and is a discrete analog for the estimate (10.14).

272

Chapter 10 Splitting of the operator at the time derivative

10.1.5 Generalizations We note some of the key research areas that focus on the synthesis and development of the above results. On the basis of the a priori estimate (10.22), we obtain the convergence of the solution of the discrete problem (10.17) to the solution of the differential problem (10.1), (10.2) with the first order by  . In the standard way, we consider the problem for the error using a particular scheme for finding the solution at the first time level. Instead of (10.17), we can employ another additive scheme. In the class of vector additive schemes, in particular, special attention should be given to the scheme ˛ X ˇ D1

Bˇ

yˇnC1  yˇn 

C

p X

Bˇ

yˇn  yˇn1

ˇ D˛C1



C A.y˛nC1 C .1  2 /y˛n C y˛n1 / D ' n , n D 0, 1, : : : ,

˛ D 1, 2, : : : , p.

In this case, the time derivative of several components of the vector solution is shifted to the upper time level. Such vector additive schemes are in common use [1,158] with the usual splitting (10.4). Some resources are available when considering more general problems than (10.1), (10.2), (10.5). In our study, we restricted ourselves to the simplest problems, where the operators A, B and the components of splitting B˛ , ˛ D 1, 2, : : : , p are constant, self-adjoint, and positive in a finite-dimensional Hilbert space H . These restrictions can be removed in some cases, by analogy with the theory of additive schemes for the problems (10.1), (10.2) with the usual splitting (10.5), considering, e.g., problems with non-self-adjoint operators, and problems with operator factors [136]. In terms of generalizing the results, of greatest interest is to construct additive operator-difference schemes for solving the Cauchy problem for the evolutionary equation (10.1) with the splitting of both the operator A and the operator B, i.e., for the problem (10.1), (10.2), (10.4), (10.5). In this case, the transition to the new time level is based on solving a sequence of problems for the equations du˛ C A˛ u˛ D f˛ .t /, dt with appropriate initial data. B˛

t >0

˛ D 1, 2, : : : , p

10.2 General splitting Many problems of practical interest are described by initial-boundary value problems for systems of time-dependent PDEs, where individual components of the vector of unknowns are coupled to each other, and therefore splitting schemes are applied to obtain a simple problem for numerically finding the solution components at a new time level.

Section 10.2 General splitting

273

Typically, additive operator-difference schemes for systems of evolutionary equations are constructed for operators interconnected in space. Here we investigate more general problems, where coupling of the time derivatives takes place for components of the solution vector. Splitting schemes are designed using an additive representation for both the leading operator of the problem and the operator at the time derivative. They are based on a triangular two-component representation of the operators.

10.2.1 Preliminary discussions As a rule, mathematical modeling of applied problems is based on numerical solving of boundary value problems for systems of time-dependent PDEs. To construct numerical algorithms for solving such problems, we apply approximations to the equations taking into account the corresponding initial data and boundary conditions. The approximation in space is conducted using finite difference schemes, finite element procedures, or finite volume methods. Special attention should be given to approximations in time for solving problems with systems of equations. In addition to general conditions for approximation and stability, we should keep in mind the question of the computational implementation of the schemes under consideration. To solve the corresponding discrete problem at a new time level in an efficient way, special additive operator-difference schemes (splitting schemes) demonstrate great advantages. In solving initial-boundary value problems for multidimensional PDEs, a transition to a chain of simpler problems allows us to construct economical difference schemes – we speak of splitting with respect to spatial variables. In some cases, it is useful to split the original problem into subproblems of a distinct nature – we have splitting into physical processes. Recently, active discussions have been concerned with regionally additive schemes (domain decomposition methods), which are oriented to parallel computers. Additive schemes for vector problems may be treated as a separate class. The schemes of this type can be used to develop efficient numerical algorithms for solving systems of time-dependent PDEs. A typical situation is the case where individual components of the unknown vector are interconnected, and it is difficult to derive a simple problem for evaluating the vector components at a new time level. Various classes of additive schemes are developed for vector problems [154]. As a rule, individual components of the vector of unknowns are interconnected in these problems. The use of splitting schemes is aimed at obtaining simple enough problems for the individual components of the solution at a new time level. For parabolic and hyperbolic systems of equations with a self-adjoint elliptic operator, locally onedimensional additive schemes are constructed in [131] using the regularization principle for difference schemes. To design efficient splitting schemes for systems of equations, the Samarskii alternating triangle method can be employed, which is generally treated as an iterative method [131, 138]. This approach is implemented, in particular, in [89] for dynamic problems of elasticity, and in [211] for problems of an incompress-

274

Chapter 10 Splitting of the operator at the time derivative

ible fluid with a variable viscosity. Additive schemes for time-dependent vector equations of first and second order are studied in [194] for problems of electrodynamics. Traditionally, additive operator-difference schemes for systems of evolutionary equations are in common use in constructing approximations to equations with operators coupled in space. In some cases, an interconnection occurs between the time derivatives of individual components of the solution vector. Therefore, it is necessary to design additive operator-difference schemes with splitting the operator at the time derivative. The theory and practice of constructing such kinds of splitting schemes are just beginning to be developed. In fact, the paper [205] is the first study on splitting schemes for problems with an additive representation of the operator at the time derivative for evolutionary equations of first order. New vector additive schemes have been proposed and investigated in this work using splitting the operator at the time derivative into the sum of positive definite and self-adjoint operators. Unfortunately, these schemes cannot be applied directly to the systems of evolutionary equations with a coupling between the time derivatives. Here we propose splitting schemes based on an additive representation for both the leading operator of the problem and the operator at the time derivative. The schemes employ a triangular two-component representation of the operators and are applied to systems of evolutionary equations. The Cauchy problem for a system of PDEs is considered as an illustrative example using various approximations in space. The emphasis is on constructing additive schemes with a triangular splitting of the leading operator of the problem. A general problem of the splitting of the operator at the time derivative is examined, too.

10.2.2 Problem formulation Let H˛ , ˛ D 1, 2, : : : , p be finite-dimensional real Hilbert (Euclidean) spaces, where the scalar product and the norm are denoted by ., /˛ and k  k˛ , ˛ D 1, 2, : : : , p, respectively. The individual components of the solution are denoted by u˛ .t /, ˛ D 1, 2, : : : , p for every t (0  t  T , T > 0). We seek the solution for the system of evolutionary equations of first order: p X ˇ D1

p X duˇ B˛ˇ A˛ˇ uˇ D f˛ , C dt

˛ D 1, 2, : : : , p.

(10.23)

ˇ D1

Here f˛ .t / 2 L2 .0, T ; H˛ /, ˛ D 1, 2, : : : , p are specified, and B˛ˇ , A˛ˇ are linear constant (independent of t ) operators acting from Hˇ onto H˛ (A˛ˇ : Hˇ ! H˛ , B˛ˇ : Hˇ ! H˛ ) for all ˛ D 1, 2, : : : , p. The system of equations (10.23) is supplemented with the initial data u˛ .0/ D v˛0 ,

˛ D 1, 2, : : : , p.

(10.24)

275

Section 10.2 General splitting

We treat the system of equations (10.23) as a single evolutionary equation for the vector u D ¹u1 , u2 , : : : , up º: B

du C Au D f .t /, dt

0 < t  T,

(10.25)

where f D ¹f1 , f2 , : : : , fp º, and the elements of the operator matrices A and B are represented in the form A D ¹A˛ˇ º,

B D ¹B˛ˇ º,

˛, ˇ D 1, 2, : : : , p.

On the direct sum of spaces [59] H D H1 ˚ H2 ˚    ˚ Hp , we put .u, v/ D

p X

.u˛ , v˛ /˛ ,

kuk2 D

˛D1

p X

ku˛ k2˛ .

˛D1

In view of (10.24), we define u.0/ D v0 ,

(10.26)

where v0 D ¹v10 , v20 , : : : , vp0 º. Consider the Cauchy problem (10.25), (10.26) under the condition that the operators A and B are self-adjoint and positive definite in H , i.e., A D A   ıA E ,

ıA > 0,

B D B   ıB E ,

ıB > 0,

(10.27)

where E is the identity operator in H . The self-adjointness is associated with the fulfillment of the equalities A˛ˇ D Aˇ ˛ ,

B˛ˇ D Bˇ˛ ,

˛, ˇ D 1, 2, : : : , p

for the operators of the original system of equations (10.23). Here is an elementary a priori estimate for the solution of the Cauchy problem (10.25), (10.26). We will use it as a guide in investigating the corresponding operatordifference schemes. For D D D  > 0, we use notation HD for a space H equipped with the scalar product .y, w/D D .Dy, w/ and the norm kykD D .Dy, y/1=2 . du Multiplying both sides of equation (10.25) scalarly in H by , we obtain dt     1d du du du , C .Au, u/ D f , . B dt dt 2 dt dt Taking into account (10.27) and using      du du 1  1 du  B , C B f ,f , f, dt dt dt 4 we derive the inequality

d 1 kuk2A  kf k2B 1 . dt 2

276

Chapter 10 Splitting of the operator at the time derivative

We get from it the following a priori estimate: Z 1 t kf . /k2B 1 d , ku.t /k2A  kv0 k2A C 2 0

(10.28)

which expresses the stability of the solution of the problem (10.25), (10.26) with respect to the initial data and the right-hand side. Systems of evolutionary equations similar to (10.23) result from approximations in space for many applied problems. Let us consider some typical examples for the problem (10.23), (10.24), not specifying the corresponding spaces of continuous and grid functions. First, we present the system of coupled parabolic equations of second order describing mass transfer in multicomponent media [46, 180]. The solution is sought in a bounded domain , u˛ .x, t /, x 2 , i.e., p X ˇ D1

p p X @uˇ X b˛ˇ .x/ div.k˛ˇ .x/ grad uˇ / C r˛ˇ .x/uˇ D f˛ .x, t /,  @t ˇ D1

x 2 ,

0 < t  T,

ˇ D1

˛ D 1, 2, : : : , p.

The coefficients r˛ˇ are associated with the reaction processes, whereas k˛ˇ describes the diffusion phenomena: main-term diffusion at ˛ D ˇ and cross-term diffusion coefficients at ˛ ¤ ˇ. For multicomponent media, we have b˛ˇ D ı˛ˇ b˛ , where ı˛ˇ is Kronecker’s delta. The second example concerns fluid motion in porous media. The governing equations for a flow in fractured porous media employ the multiple porosity model (see, e.g., [12, 22]). In this case, u˛ .x, t / is the dynamic pore pressure in the p-porosity model. For these models, it is the principal moment that b˛ˇ ¤ ı˛ˇ b˛ and r˛ˇ ¤ 0 (the Barenblatt model).

10.2.3 Scheme with weights To solve numerically operator-differential problem (10.25), (10.26), we use the standard scheme with weights. Introduce a uniform grid in time !  D ! [ ¹T º D ¹t n D n ,

n D 0, 1, : : : , N0 ,

N0 D T º

and, by definition, put y n D y.t n /, t n D n . Let us approximate equation (10.25) by the two-level difference scheme B

y nC1  y n C A. y nC1 C .1   /y n / D 'n , 

(10.29)

where  is a numerical parameter (weight) within 0    1, and, e.g., 'n D f . t nC1 C .1   /t n /. For simplicity, we restrict ourselves to the case of the same

277

Section 10.2 General splitting

weight for all equations in the system (10.23). In view of (10.26), we supplement (10.29) with the initial data (10.30) y 0 D v0 . A detailed study of the scheme with weights (the necessary and sufficient condition for stability as well as the choice of a norm) was conducted in [131,134]. Here we restrict ourselves to an elementary estimate for stability of the operator-difference scheme (10.29), (10.30). The estimate (10.28) serves us as a guide in our study. Theorem 10.3. If   1=2, then the operator-difference scheme .10.29/ is absolutely stable in HA , and the difference solution satisfies the levelwise estimate  (10.31) ky nC1 k2A  ky n k2A C k'n k2 1 . .BC. 12 /A/ 2 Proof. Write the scheme (10.29) in the form    y nC1  y n 1 y nC1 C y n BC  A CA D 'n . 2  2 Multiplying both sides of this equation scalarly in H by 2.y nC1  y n /, we obtain the inequality    y nC1  y n y nC1  y n  1 2 B C   A , 2    nC1  y n  nC1 nC1 n n n y C .Ay ,y /  .Ay , y / D 2 ' , .  Using the inequality     y nC1  y n y nC1  y n  nC1  y n  1 n y  BC  A , ' ,  2        1 1 1 BC  'n , 'n , C A 4 2 we derive the required estimate (10.31). The estimate (10.31) is just a discrete analog of the estimate (10.28) and ensures the unconditional stability of the difference scheme with weights (10.29), (10.30) under the natural conditions   1=2. Considering the corresponding problem for the error, we prove the convergence of the solution of the operator-difference problem (10.25), (10.26) to the solution of the operator-differential problem (10.25), (10.26) in HA under the restriction   1=2 O..2  1/ C  2 /. If  D 1=2, then we have the second-order convergence rate with respect to  . The operator-difference scheme (10.29) may be written in the canonical form for the two-level scheme: .B C   A/

y nC1  y n C Ay n D 'n . 

(10.32)

278

Chapter 10 Splitting of the operator at the time derivative

The transition to a new time level requires to solve the problem .B C   A/y nC1 D

n

.

Concerning the original problem (10.23), (10.24), we must solve the system of coupled equations p X .B˛ˇ C  A˛ˇ /yˇnC1 D ˛n , ˛ D 1, 2, : : : , p. ˇ D1

Various iterative methods can be used for this procedure [114, 138]. Another opportunity is to take into account the specific features of the above unsteady problems and to construct splitting schemes in which the transition to a new time level involves the solution of simpler problems. For the problems of type (10.23), (10.24), it seems reasonable to employ the splitting schemes where the transition to a new time level is performed via solving the problems .B˛˛ C  A˛˛ /y˛nC1 D en˛ ,

˛ D 1, 2, : : : , p.

This means that we have to invert only the diagonal part of the operator matrix B C   A in our computations.

10.2.4 Schemes with a diagonal operator We start with the case where the problem of inversion of the operator B does not exist. Such a situation occurs if the operator matrix B at the time derivatives is diagonal, i.e., B˛ˇ D ı˛ˇ B˛ ,

˛ D 1, 2, : : : , p.

(10.33)

This class of problems appears in simulations of mass transfer in multicomponent media. In this case, the components of the solution vector are coupled due to the elements A˛ˇ , ˛ ¤ ˇ in the operator matrix A. Let us construct additive operator-difference schemes using the triangular splitting of the operator A, i.e., (10.34) A D A1 C A2 , A1 D A2 . In the additive representation (10.34), we have 0 0 1 1 1 A11 0  0 2 2 A11 B B C 1 B A21 B 0 C 2 A22    C , A2 D B 0 A1 D B B  B  C   0 A @ @ 1 0 Ap1 Ap2    2 App

A12 1 2 A22  0

1    A1p C    A2p C C.    Ap1p C A 1    2 App

Instead of (10.32), we use the scheme y BQ

 yn C Ay n D 'n , 

nC1

(10.35)

279

Section 10.2 General splitting

where the operator BQ has the following factorized form: e D .B C   A1 /B 1 .B C   A2 /. B

(10.36)

The scheme (10.35), (10.36) is an operator-matrix analog of the Samarskii alternating triangle method [131]. Taking into account (10.34), due to self-adjointness and positive definiteness of B at   1=2, we have BQ D B C   A C  2  2 A1 B 1 A2 ,

BQ D BQ   B C   A.

Similarly to Theorem 10.3, we prove the following statement. Theorem 10.4. If   1=2, then the factorized operator-difference scheme .10.34/– .10.36/ is absolutely stable in HA , and the difference solution satisfies the levelwise estimate  ky nC1 k2A  ky n k2A C k'n k2 1 . .BC. 12 /AC 2  2 A1 B 1 A2 / 2 The computational implementation of the scheme (10.34)–(10.36) for solving the problem (10.23), (10.24), (10.33) can be arranged using the sequence of simpler problems:    B˛ C  A˛˛ y˛nC1=2 D L ˛n , 2    B˛ C  A˛˛ y˛nC1 D O ˛n , ˛ D 1, 2, : : : , p. 2 Similarly to the scheme with weights (10.29), the factorized scheme (10.34)–(10.36) has convergence of second order at  D 1=2 and first-order convergence for any other values of the weights.

10.2.5 The more general problem If the problem (10.25), (10.26) has a nondiagonal operator B (B˛ˇ ¤ ı˛ˇ B˛ ), then additive operator-difference schemes can be constructed using the triangular splitting for both the operator A and the operator B. Similarly (10.34), assume that B D B 1 C B2 ,

B1 D B2 .

(10.37)

We focus on the scheme with weights (10.32) that, in view of (10.34), (10.37), may be written in the form y nC1  y n (10.38) C Ay n D 'n , C  where C D C 1 C C2 ,

C1 D B1 C   A1 ,

C2 D B2 C   A2 .

(10.39)

280

Chapter 10 Splitting of the operator at the time derivative

By (10.27), we have C1 D C2 ,

1 C˛  .ıB C   ıA /E . 2

The operator C1 C C2 may be represented as C1 C C2 D

1 1 .C1 C "E /.C2 C "E /  .C1  "E /.C2  "E / 2" 2"

with arbitrary " > 0. The value of " will be specified a little bit later. Instead of the two-level scheme (10.38), we employ the three-level scheme y nC1  y n 1 .C1 C "E /.C2 C "E / 2"  y n  y n1 1 C Ay n D 'n .  .C1  "E /.C2  "E / 2" 

(10.40)

The primary potential advantage of this scheme, in comparison with the scheme (10.38), is that its implementation is based on the inversion of the factorized operator .C1 C "E /.C2 C "E / at the new time level. Taking into account that y nC1  y n1  y nC1  2y n C y n1 y nC1  y n , D C  2 2 2 y nC1  y n1  y nC1  2y n C y n1 y n  y n1 , D   2 2 2 we can write the scheme (10.40) as follows: C

y nC1  y n1 y nC1  2y n C y n1 C Ay n D 'n , CD 2 2

where

(10.41)

 .C1 C2 C "2 E /. 2" In view of C D B C   A, we verify directly that the operator-difference scheme (10.41) approximates equation (10.25) with first-order accuracy with respect to  for each " D O.1/. We now formulate the sufficient conditions for the stability of this scheme. An exhaustive study of stability of three-level schemes with self-adjoint operators was done in [131, 134, 136]. We do not use here the general results on the stability of operator-difference schemes from the above works. Similar to [20], we obtain immediately elementary estimates for stability with respect to the initial data and the right-hand side. Taking into account DD

1 1 y n D .y nC1 C 2y n C y n1 /  .y nC1  2y n C y n1 / 4 4

281

Section 10.2 General splitting

we write (10.41) as

  nC1 2 y y nC1  y n1  2y n C y n1 C D A C 2 4 2 nC1 n n1 y  2y C y CA D 'n . 4

Introducing

1 vn D .y n C y n1 /, 2 we can rewrite (10.42) in the form C

wn D

y n  y n1 , 

wnC1  wn 1 wnC1 C wn CR C A.vnC1 C y n / D 'n , 2  2

where RDD

(10.42)

(10.43)

2 A. 4

Multiplying scalarly both sides of (10.43) by 2.vnC1  vn / D  .wnC1 C wn /, we get the equality  .C .wnC1 C wn /, wnC1 C wn / C .R.wnC1  wn /, wnC1 C wn / 2 C .A.vnC1 C vn /, vnC1  vn / D  .'n , wnC1 C wn /. For the right-hand side we use the estimate 1 1 .'n , wnC1 C wn /  .C .wnC1 C wn / C .C 1 'n , 'n /. 2 2 This yields from (10.44) the inequality  E nC1  E n C .C 1 'n , 'n /, 2 where we use the notation

(10.44)

(10.45)

E n D .Avn , vn / C .Rwn , wn /. Inequality (10.45) will be the desired a priori estimate, if we show that E n defines the squared norm of the difference solution. By the positivity of A, it is sufficient to require the non-negativity of the operator R. With the above-mentioned notation we have  2  .C1 C "E /.C2 C "E / C .C1  "E /.C2  "E /  A 4" 4" 4 2 2   .C1 C "E /.C2 C "E /  A > .  1/A. 4" 4 4

RD

Thus, R > 0 for   1. The result of our considerations is the following statement.

282

Chapter 10 Splitting of the operator at the time derivative

Theorem 10.5. If   1, then the operator-difference scheme .10.34/, .10.37/, .10.40/ is absolutely stable, and the difference solution satisfies a priori estimate .10.45/ with  n  n    y C y n1 2  y  y n1 2 n     . E D  C  2  A R The proven estimate (10.45) ensures the stability of operator-difference scheme (10.34), (10.37), (10.40) with respect to the initial data and the right-hand side. It can be regarded as a more complex analog of (10.31) and it agrees with the estimate (10.28) for the solution of the problem (10.25), (10.26). The computational realization of the scheme (10.40) involves the solution of the grid problem (10.46) .C1 C "E /.C2 C "E /y nC1 D n at the new time level. Introducing the auxiliary unknown y nC1=2 , we have for (10.46) the following representation: .C1 C "E /y nC1=2 D .C2 C "E /y

nC1

Dy

n

,

nC1=2

.

Taking into account the triangular structure of the operators C1 and C2 , we consistently solve the problem .B˛˛ C  A˛˛ C 2"E˛ /y˛nC1=2 D L ˛n , .B˛˛ C  A˛˛ C 2"E˛ /y˛nC1 D O ˛n , ˛ D 1, 2, : : : , p, where E˛ is the identity operator in H˛ .

10.3 Explicit-implicit splitting schemes In many applied problems, the individual components of the unknown vector are interconnected and therefore splitting schemes are applied in order to get a simple problem for evaluating unknowns at a new time level. On the basis of additive schemes (splitting schemes), efficient computational algorithms are constructed for numerically solving the initial value problems for systems of time-dependent PDEs. The present part of the book deals with computational algorithms that are based on using explicitimplicit approximations in time. Typically, additive operator-difference schemes for systems of evolutionary equations are constructed for operators that are coupled in space. Here we investigate more general problems, where we have the coupling of derivatives in time for components of the solution vector.

10.3.1 Introduction The above-mentioned classes of additive operator-difference schemes for evolutionary equations are based on an additive splitting of the leading operator into several terms.

283

Section 10.3 Explicit-implicit splitting schemes

For many problems of practical interest, it is interesting to investigate the problems that have an additive representation for the operator at the time derivative. In the first publication on this subject [205], vector additive operator-difference schemes were proposed and examined, where the operator at the time derivative was split into the sum of self-adjoint and positive definite operators. Among additive schemes, we highlight explicit-implicit schemes, where the different nature of terms of the problem operator is taken into account via inhomogeneous approximations in time. Explicit-implicit schemes are widely used for the numerical solution of convection-diffusion problems. Various variants of inhomogeneous discretization in time are given in [7]. One or another explicit approximation is applied to the convective transport operator, whereas the diffusive transport operator is approximated implicitly. Thus, the most severe restrictions on a time step due to diffusion are removed. In view of the subordination of the convective transport operator to the diffusive transport operator, we have already proved unconditional stability of the above-considered explicit-implicit schemes for time-dependent convection-diffusion problems. Similar techniques are used in the analysis of diffusion-reaction problems. In this case (see, e.g., [113]), diffusive transport is treated implicitly, whereas for reactions (source terms), explicit approximations are used. Such explicit approximations demonstrate obvious advantages for problems with nonlinear terms describing reaction processes. Here we propose splitting schemes for additive representation of the leading operator of the problem, i.e., the operator at the time derivative. We separate the diagonal part of a problem operator matrix and employ explicit-implicit approximations in time.

10.3.2 Boundary value problems for systems of equations We consider the boundary value problem for the system of coupled parabolic equations in a bounded domain . For the individual components u˛ .x, t /, x 2 , ˛ D 1, 2, : : : , p, we have p X ˇ D1

c˛ˇ .x/

p @uˇ X div.k˛ˇ .x/ grad uˇ / D f˛ .x, t /,  @t

x 2 .

(10.47)

ˇ D1

The system of equations (10.47) is supplemented with the following boundary and initial conditions, respectively: u˛ .x, t / D 0, u˛ .x, 0/ D

x 2 @,

u0˛ .x/,

0 < t  T,

x 2 ,

˛ D 1, 2, : : : , p.

(10.48) (10.49)

We formulate the main restrictions on the coefficients for problem (10.47)–(10.49). The system of parabolic equations is considered under the restrictions p p X X c˛ˇ ˛ ˇ  ı ˛2 , ı > 0, c˛ˇ D cˇ ˛ , ˛,ˇ D1

˛D1

284

Chapter 10 Splitting of the operator at the time derivative p X

k˛ˇ D kˇ ˛ ,

k˛ˇ ˛ ˇ 

p X

˛2 ,

> 0.

˛D1

˛,ˇ D1

For real matrices C D ¹c˛ˇ º and K D ¹k˛ˇ º, we have C D C   ıI ,

K D K   I ,

where I is the p p identity matrix. After approximation in space, from problem (10.47) with boundary conditions (10.48), we arrive at a system of ODEs. Let us formulate the corresponding Cauchy problem. Let H˛ , ˛ D 1, 2, : : : , p be finite-dimensional real Hilbert (Euclidean) spaces of grid functions, where the scalar product and the norm are denoted by ., /˛ and k  k˛ , ˛ D 1, 2, : : : , p, respectively. The individual components of the solution are denoted by u˛ .t / 2 H˛ , ˛ D 1, 2, : : : , p for every t (0  t  T , T > 0). We seek the solution for the system of evolutionary equations of first order: p X ˇ D1

B˛ˇ

p X duˇ A˛ˇ uˇ D f˛ , C dt

˛ D 1, 2, : : : , p.

(10.50)

ˇ D1

Here f˛ .t / 2 L2 .0, T ; H˛ /, ˛ D 1, 2, : : : , p are specified, and B˛ˇ , A˛ˇ are linear constant (independent of t ) operators acting from Hˇ onto H˛ (A˛ˇ : Hˇ ! H˛ , B˛ˇ : Hˇ ! H˛ ) for all ˛, ˇ D 1, 2, : : : , p. The system of equations (10.50) is supplemented with the initial data u˛ .0/ D v˛0 ,

˛ D 1, 2, : : : , p.

(10.51)

We treat the system of equations (10.50) as a single evolutionary equation for vector u D ¹u1 , u2 , : : : , up º: B

du C Au D f .t /, dt

0 < t  T,

(10.52)

where f D ¹f1 , f2 , : : : , fp º, and the elements of the operator matrices A and B are represented in the form A D ¹A˛ˇ º,

B D ¹B˛ˇ º,

˛, ˇ D 1, 2, : : : , p.

On the direct sum of spaces [59] H D H1 ˚ H2 ˚    ˚ Hp , we put .u, v/ D

p X

.u˛ , v˛ /˛ ,

kuk2 D

˛D1

p X

ku˛ k2˛ .

˛D1

In view of (10.51), we define u.0/ D v0 , where v0 D ¹v10 , v20 , : : : , vp0 º.

(10.53)

285

Section 10.3 Explicit-implicit splitting schemes

Consider Cauchy problem (10.52), (10.53) under the condition that operators A and B are self-adjoint and positive definite in H , i.e., A D A   ıA E ,

ıA > 0,

B D B   ıB E ,

ıB > 0,

(10.54)

where E is the identity operator in H . The self-adjointness is associated with the fulfillment of the equalities A˛ˇ D Aˇ ˛ ,

B˛ˇ D Bˇ˛ ,

˛, ˇ D 1, 2, : : : , p

for the operators of the original system of equations (10.50). Here is an elementary a priori estimate for the solution of Cauchy problem (10.52), (10.53). We will use it as a guide in investigating the corresponding operator-difference schemes. For D D D  > 0, we use notation HD for a space H equipped with the scalar product .y, w/D D .Dy, w/ and the norm kykD D .Dy, y/1=2 . We have the following a priori estimate: Z 1 t 2 0 2 kf . /k2B 1 d , (10.55) ku.t /kA  kv kA C 2 0 which expresses the stability of the solution of problem (10.52), (10.53) with respect to the initial data and the right-hand side.

10.3.3 Schemes with a diagonal operator To solve numerically the operator-differential problem (10.52), (10.53), we use the standard scheme with weights. We introduce a uniform grid in time !  D ! [ ¹T º D ¹tn D n ,

n D 0, 1, : : : , N0 ,

N0 D T º

and, we denote y n D y.t n /, t n D n . We start with the case where the problem of inversion of the operator B does not exist. Such a situation occurs if the operator matrix B is diagonal at the time derivatives, i.e., (10.56) B˛ˇ D ı˛ˇ B˛ , ˛ D 1, 2, : : : , p. This class of problems appears in simulations of mass transfer in multicomponent media. In this case, the components of the solution vector are coupled due to the elements A˛ˇ , ˛ ¤ ˇ in the operator matrix A. Let us construct additive operator-difference schemes using the splitting of operator A with separation of the diagonal part. In this case, we obtain A D A0 C A1 ,

A0 D diag.A11 , A22 , : : : , App /.

In additive representation (10.57), we have 0 0 1 A11 0    0 0 A12 B 0 A22    0 C B A21 0 B C A0 D B @             A , A1 D @       Ap1 Ap2 0 0    App

1    A1p    A2p C C.   A  0

(10.57)

286

Chapter 10 Splitting of the operator at the time derivative

In our problem, in view of (10.54), we get A0 C A1  ıA E ,

ıA > 0.

(10.58)

Let us consider problem (10.50), (10.51) under the additional assumption: A0  A1  0.

(10.59)

In some cases, property (10.59) follows from (10.58). The properties of operator A are associated with the properties of matrix K, i.e., with the coefficients k˛ˇ , ˛, ˇ D 1, 2, : : : , p in the boundary value problem (10.47)– (10.49). The positive definiteness of operator A follows from the positive definiteness of matrix K. In view of (10.57), the fulflilment of (10.59) may be associated with eDK e   0, K where

e k ˛˛ D k˛˛ ,

e D ¹e K k ˛ˇ º,

e k ˛ˇ D k˛ˇ ,

˛, ˇ D 1, 2, : : : , p.

For the system of two equations (p D 2), from A0 C A1  0, (10.59) follows immediately. For p D 3, we can highlight the case with non-positive off-diagonal coefficients: k˛ˇ  0, ˛, ˇ D 1, 2, 3, ˛ ¤ ˇ. e D ¹jk˛ˇ jº  0 [95], which ensures the fulfillment Under these restrictions, we have K of (10.59). For general systems, we emphasize the case of diagonal dominance of matrix K, where p X jk˛ˇ j. k˛˛  ˛¤ˇ D1

These examples demonstrate the fulfillment of conditions (10.58), (10.59) in a number of problems with the decomposition (10.57). For numerically solving (10.52), (10.53) under constraints (10.56)–(10.58), we will use the two-level scheme: y nC1  y n (10.60) C A0 y nC1 C A1 y n D 'n .  This scheme with inhomogeneous approximation in time belongs to the class of explicit-implicit schemes. Here only the diagonal part of the operator A is shifted to the upper time level. The computational implementation of the explicit-implicit scheme (10.60) is conducted by means of problems B

.B˛˛ C A˛˛ /y˛nC1 D

n ˛,

˛ D 1, 2, : : : , p

at the new time level. For these individual problems, it is possible to arrange independent (parallel) computing y˛nC1 , ˛ D 1, 2, : : : , p. The main result on stability of the explicit-implicit scheme is formulated as the following statement.

Section 10.3 Explicit-implicit splitting schemes

287

Theorem 10.6. If .10.59/ holds, then an explicit-implicit difference scheme satisfying .10.54/, .10.57/, .10.60/ is unconditionally stable, and for the difference solution the following levelwise estimate is valid:  ky nC1 k2A  ky n k2A C k'n k2  1 . .BC 2 .A0 A1 // 2

(10.61)

Proof. For the proof, we write the explicit-implicit scheme in the form  nC1  y y nC1 C y n  yn  CA D 'n B C .A0  A1 / 2  2 and we multiply it scalarly in H by 2.y nC1  y n /. Further arguments are similar to the proof of Theorem 10.3. Estimate (10.61) for stability with respect to the initial data and the right-hand side, proven for the explicit-implicit scheme (10.60), is not significantly different from the estimate corresponding to the standard scheme with weights (see (10.31)). Explicit-implicit scheme (10.60) approximates equation (10.52) with the first order with respect to  . It is possible to construct unconditionally stable explicit-implicit schemes with a second-order approximation in time. Instead of (10.60), we can use the three-level explicit-implicit scheme: B

y nC1  y n1 C A0 . y nC1 C .1  2 /y n C  y n1 / C A1 y n D 'n 2

(10.62)

with 'n D f .t n /. To calculate the first step, we can apply, e.g, the two-level scheme y1  y0 y1 C y0 '1 C '0 C .A0 C A1 / D .  2 2 The investigation of stability is based on the following general statement from the theory of stability (correctness) for three-level operator-difference schemes [131,134, 136]. B

Lemma 10.1. Let in the three-level operator-difference scheme B

y nC1  2y n C y n1 y nC1  y n1 C Ay n D 'n CD 2 2

(10.63)

operators A, B, D be constant (independent of n) and A D A  > 0, B D B  > 0, D D D  > 0.

(10.64)

If D>

2 A, 4

(10.65)

288

Chapter 10 Splitting of the operator at the time derivative

then scheme .10.63/, .10.64/ is unconditionally stable and its solution satisfies the estimate  E nC1  E n C .B 1 'n , 'n /, (10.66) 2 where  n  n    y C y n1 2  y  y n1 2 n     E D  C  2 . 2  A D A 4

Proof. Taking into account 1 1 y n D .y nC1 C 2y n C y n1 /  .y nC1  2y n C y n1 /, 4 4 we write (10.63) as   nC1 y nC1  y n1 2 y  2y n C y n1 B C D A 2 4 2 y nC1  2y n C y n1 CA D 'n . 4 Let

1 vn D .y n C y n1 /, 2 and rewrite (10.67) in the form B

wn D

y n  y n1 

wnC1 C wn wnC1  wn 1 CR C A.vnC1 C vn / D 'n , 2  2

where RDD

(10.67)

(10.68)

2 A. 4

Multiplying scalarly both sides of (10.68) by 2.vnC1  vn / D  .wnC1 C wn /, we get the equality  .B.wnC1 C wn /, wnC1 C wn / C .R.wnC1  wn /, wnC1 C wn / 2 C .A.vnC1 C vn /, vnC1  vn / D  .'n , wnC1 C wn /. For the right-hand side, we use the estimate 1 1 .'n , wnC1 C wn /  .B.wnC1 C wn / C .B 1 'n , 'n /. 2 2 This makes it possible to get from (10.69) the inequality  E nC1  E n C .B 1 'n , 'n /, 2

(10.69)

(10.70)

289

Section 10.3 Explicit-implicit splitting schemes

where we use the notation E n D .Avn , vn / C .Rwn , wn /. Inequality (10.70) is the desired a priori estimate (10.66), if we show that E n defines the squared norm of the difference solution. In view of the positivity of A, it is sufficient to require the positivity for the operator R (see (10.65)). Scheme (10.62) may be written in the form (10.63) with D D   2 A0 . Taking this fact into account, stability condition (10.65) takes the form   1 0 < 4 A0  A D 4   A0 C A0  A1 . 2 Under assumptions (10.57), (10.59), this condition will be true for  > 1=2. Thus, we can formulate the following statement. Theorem 10.7. An explicit-implicit scheme .10.62/ satisfying .10.54/, .10.57/ and .10.59/ is unconditionally stable for  > 1=2, and the difference solution satisfies levelwise estimate .10.66/, where  n    n n1 2  y C y n1 2  n 2 y  y    . E D  C   2  A A0  1 A 4

10.3.4 General case For problem (10.52), (10.53) with a common (not diagonal) operator B (B˛ˇ ¤ ı˛ˇ B˛ ), explicit-implicit difference schemes will be based on a decomposition of operator B. Similarly to (10.57), we set B D B0 C B1 ,

B0 D diag.B11 , B22 , : : : , Bpp /.

(10.71)

In addition to the positive definiteness of operator B, we assume that the inequality B 0  B1  0

(10.72)

holds. Thus, coefficients c˛ˇ , ˛, ˇ D 1, 2, : : : , p of the matrix C in problem (10.47)– (10.49) are considered under restrictions similar to those formulated above for the coefficients k˛ˇ , ˛, ˇ D 1, 2, : : : , p of matrix K. To solve numerically (10.52), (10.53) under the constraints (10.54), (10.57), (10.59), (10.71), (10.72), we will use the three-level scheme: B0

y nC1  y n y n  y n1 C B1   nC1 C A0 . y C .1  2 /y n C  y n1 / C A1 y n D 'n .

Let us formulate the stability condition for this explicit-implicit scheme.

(10.73)

290

Chapter 10 Splitting of the operator at the time derivative

Theorem 10.8. An explicit-implicit difference scheme .10.72/ satisfying .10.54/, .10.57/, .10.59/, .10.71/, .10.72/ is unconditionally stable for  > 1=2, and the difference solution satisfies the levelwise estimate .10.66/, where  n  n   n1 2   y C y n1 2 n 2 y  y    . E D  C  1 2  .B0 B1 /CA0  1 A A 2

4

Proof. We employ Lemma 10.1. Scheme (10.72) may be written in the form (10.63) with  D D .B0  B1 / C   2 A0 . 2 In the terms of this theorem, stability condition (10.65) will be valid for  > 1=2.

Chapter 11

Systems of evolutionary equations with pairwise adjoint operators As usual, applied problems of mathematical physics are formulated as systems of coupled PDEs, where components of the solution are involved in several equations. That is why we need to construct additive schemes for such problems using splitting that is focused on obtaining simpler problems for finding an approximate solution via standard problems formulated for the individual components of the solution. We emphasize a class of systems of evolutionary equations that is characterized by the presence of adjoint operators in various equations of the system. Such a structure is typical for mathematical models describing many applied problems. In particular, we note problems of continuum mechanics: dynamics of a compressible fluid, acoustics, thermoelasticity, consolidation in porous media flows and so on. In this regard, particular attention should be given to problems of electrodynamics, where we have the system of Maxwell equations for electric and magnetic fields. For systems of evolutionary equations with self-adjoint operators, we construct the classical schemes with weights as well as additive schemes based on splitting into simpler problems for individual components of the solution.

11.1 Splitting schemes for a system of equations with pairwise adjoint operators Now we consider finite difference approximations in time for numerically solving the Cauchy problem formulated for a special system of evolutionary equations of first order that includes pairwise adjoint operators. Such problems appear, e.g., after approximation in space for a parabolic equation of second order, where the solution itself and its first derivatives with respect to spatial variables are treated as unknowns. We construct two-level operator-difference schemes with weights both for the original and for the transformed system of evolutionary equations. Stability conditions for these operator-difference schemes are obtained in the corresponding Hilbert spaces. The emphasis is on the construction and investigation of additive schemes (splitting schemes) that are associated with the solution of elementary problems at every time step.

292

Chapter 11 Equations with pairwise adjoint operators

11.1.1 Preliminary discussions In solving applied problems, we deal with boundary value problems for systems of time-dependent PDEs. To construct computational algorithms for such problems, we approximate equations taking into account appropriate initial and boundary conditions. Approximation in space is based on finite difference schemes, finite element procedures or finite volume methods [51, 70, 115, 131]. Special requirements are applied to approximation in time for numerical solving of problems for systems of equations [6, 55, 88]. In addition to the general requirements to satisfy the conditions of approximation and stability, it is necessary to keep in mind the issues of computational implementation of the constructed schemes, i.e., the issue of how to solve the corresponding discrete problem at a new time level. In this regard, the most impressive results are associated with the construction of special additive operator-difference schemes (splitting schemes) [93, 151]. Additive schemes (operator-splitting schemes) are used to solve various unsteady problems [93, 131, 151, 218]. They are designed for efficient computational implementation of the corresponding discrete problem defining the approximate solution at a new time level. The transition to a chain of simpler problems allows us to construct efficient difference schemes – we speak of splitting with respect to spatial variables (locally one-dimensional schemes). In some cases, it is useful to separate subproblems of a distinct nature – we have splitting into physical processes. Regionally additive schemes (domain decomposition methods) are focused on constructing computational algorithms for parallel computers. The main theoretical results on stability and convergence of additive schemes have been obtained for scalar evolutionary first-order equations and, in some cases, for second-order equations [93, 131, 151, 218]. In computational practice, it is essential to construct splitting schemes for systems of evolutionary equations. For example, vector problems have individual components of the unknown vector that are interconnected with each other. In this case, the use of appropriate splitting schemes is intended to obtaining simple problems for the individual components of the solution at a new time level. For standard parabolic and hyperbolic systems of equations with a self-adjoint elliptic operator, additive schemes have been constructed in [131] using the regularization principle for difference schemes. Splitting schemes for systems of equations can be constructed employing the triangular splitting of a problem operator into the sum of operators adjoint to each other, i.e., using the alternating triangle method developed by Samarskii. Additive schemes of this type were used in [89] for dynamic problems of elasticity. A similar approach [154,211] was applied to problems of an incompressible fluid with a variable viscosity. Additive schemes for transient problems of electrodynamics were considered in [194]. Mathematical models of many applied problems [183] have a common structure that is characterized by the presence of pairwise adjoint operators. In our papers [200,206],

Section 11.1 Splitting schemes for a system of equations

293

we have considered the Cauchy problem for a special linear system of first-order equations with adjoint operators in a Hilbert space. Such a system of equations arises from approximation in space, e.g., for acoustic problems (compressible fluid dynamics) and electrodynamics. Another system of equations, which is typical, in particular, for incompressible fluid dynamics problems, has been considered in the work [199]. Here we construct and study the standard two-level schemes with weights, and then we analyze issues of computational implementation of these schemes. The main result is in constructing splitting schemes that are based on solving separate problems for individual operators. Systems of second-order evolutionary equations discussed in [200, 206] can be reduced to a second-order evolutionary equation for a single component of the solution. Here we consider a system that can be transformed to a single equation of first order with a self-adjoint operator. Such a structure appears if we use hybrid and mixed finite elements [17, 112] for numerically solving boundary value problems for a second-order parabolic equation. In this case, we use as the solution unknowns the solution function itself and its first derivatives with respect to spatial variables. We construct two-level operator-difference schemes both for the original system and the transformed system of equations, obtaining estimates for stability in the corresponding Hilbert spaces. The primary object of our study consists in constructing unconditionally stable additive schemes that are implemented via the successive solution of elementary problems at every time step.

11.1.2 Statement of the problem Let H˛ , ˛ D 1, 2, : : : , p be real finite-dimensional Hilbert (Euclidean) spaces, where the scalar product and norm are ., /˛ and k  k˛ , ˛ D 1, 2, : : : , p, respectively. The individual components of the solution are denoted by u˛ .t /, ˛ D 1, 2, : : : , p for every t (0  t  T , T > 0). We seek the solution to the system of equations u˛ C A˛ up D f˛ ,

˛ D 1, 2, : : : , p  1,

dup X  A˛ u˛ D fp ,  dt ˛D1

(11.1)

p1

0 < t  T.

(11.2)

Here f˛ .t / 2 L2 .0, T ; H˛ /, ˛ D 1, 2, : : : , p are prescribed, and A˛ stands for a linear constant (independent of t ) operator acting from Hp onto H˛ for all ˛ D 1, 2, : : : , p  1. The system of equations (11.1), (11.2) is supplemented with the initial condition up .0/ D vp0

(11.3)

for the last component of the solution. Other components at the initial time moment are specified by vp0 . In accordance with (11.1), (11.2), we put u˛ .0/ D v˛0 ,

v˛0 D A˛ vp0 C f˛ .0/,

˛ D 1, 2, : : : , p  1.

(11.4)

294

Chapter 11 Equations with pairwise adjoint operators

The problem (11.1)–(11.4) appears as the result of approximation in space during numerical solving of boundary value problems for PDEs. As a typical example, we mention a boundary value problem for a parabolic equation of second order. In a bounded domain , an unknown function u.x, t /, x D .x1 , x2 , : : : , xm / satisfies the equation   m @u @u X @  k.x/ D f .x, t /, x 2 , 0 < t  T , (11.5) @t ˛D1 @x˛ @x˛ where k.x/  > 0, x 2 . The equation (11.5) is supplemented with the homogeneous Dirichlet boundary conditions u.x, t / D 0,

x 2 @,

0 < t  T.

(11.6)

In addition, we specify the initial condition x 2 .

u.x, 0/ D u0 .x/,

(11.7)

We define the differential operators acting in particular directions x˛ , ˛ D 1, 2, : : : , m via the relations A˛ u D k 1=2 .x/

@u , @x˛

˛ D 1, 2, : : : , m.

(11.8)

On the set of functions that satisfy the boundary conditions (11.6), in H D L2 ./, for the adjoint operators, we have @.k 1=2 .x/v/ , ˛ D 1, 2, : : : , m. @x˛ In view of (11.8), (11.9), we get   X m m X @u @ A˛ A˛ u.  k.x/ D @x @x ˛ ˛ ˛D1 ˛D1 A˛ v D 

(11.9)

This makes it possible to write the problem (11.5)–(11.7) in the form of the Cauchy problem for the operator-differential equation du X  A˛ A˛ u D f .t /, C dt ˛D1 m

0 < t  T,

u.0/ D u0 .

(11.10) (11.11)

The equation (11.10) may be written as a system. Suppose, e.g., q˛ C A˛ u D 0, then

˛ D 1, 2, : : : , m,

du X  A q˛ D f .t /,  dt ˛D1 ˛

(11.12)

m

0 < t  T.

(11.13)

Section 11.1 Splitting schemes for a system of equations

295

In the problem (11.11)–(11.13), the unknown quantities are the solution u itself and its first derivatives q˛ , ˛ D 1, 2, : : : , m (see (11.8)). After approximation in space, from (11.11)–(11.13), we obtain our problem (11.1)– (11.3) (m D p  1, u˛ D q˛ , ˛ D 1, 2, : : : , m, up D u). Details of the choice of specific approximations and the corresponding finite-dimensional spaces are omitted here. Note only that in approximating a differential problem in space, we must preserve the important properties of adjointness for discrete operators. Instead of (11.11)–(11.13), we can use a slightly different system of equations, where du dq˛ C A˛ D 0, ˛ D 1, 2, : : : , m, dt dt m X A˛ q˛ D f .t /, 0 < t  T . u ˛D1

This corresponds to the situation, where, instead of (11.1), (11.2), we consider the equations dup du˛ C A˛ D f˛ , dt dt p1 X A˛ u˛ D fp , up 

˛ D 1, 2, : : : , p  1,

(11.14)

0 < t  T,

(11.15)

˛D1

and we solve the Cauchy problem (11.4), (11.14), (11.15).

11.1.3 A priori estimates Let us present elementary a priori estimates for the solution of the problem (11.1)– (11.2), which will be used in the study on stability of operator-difference schemes. In this case, we employ the possibility of transforming the original system of equations (11.1)–(11.2). First, we can eliminate the variables u˛ , ˛ D 1, 2, : : : , p  1 from the system (11.1)–(11.2). Substituting (11.1) into (11.2), we arrive at the evolutionary equation for up : p1 X dup ep , 0 < t  T , A˛ A˛ up D f (11.16) C dt ˛D1 where ep D fp C f

p1 X

A˛ f˛ .

˛D1

Multiply equation (11.16) scalarly in Hp by up . Taking into account A˛ A˛  0,

˛ D 1, 2, : : : , p  1,

296

Chapter 11 Equations with pairwise adjoint operators

we obtain the inequality By

1d ep , up /p . kup kp2  .f 2 dt

(11.17)

ep , up /p  kf ep kp kup kp , .f

from (11.17), we have

d ep kp . kup kp  kf dt In view of Gronwall’s lemma and the initial condition (11.3), we obtain the desired estimate Z t 0 ep . /kp d , kup k  kv kp C kf (11.18) p

0

which expresses the stability of the solution of the problem (11.3), (11.16) with respect to the initial data and the right-hand side. The second (and less trivial) approach to the transformation of the system (11.1), (11.2) is associated with the elimination of the required variable up . This leads us to the system of evolutionary equations p1 X du˛ e˛ , Aˇ uˇ D f C A˛ dt

0 < t  T,

(11.19)

ˇ D1

where

e˛ D df˛  A˛ fp , ˛ D 1, 2, : : : , p  1. f dt The equations (11.19) are supplemented with the initial conditions (11.4). Multiplying each equation from the system (11.19) scalarly in H˛ by u˛ and adding them, we obtain  p1 p1 p1 X  p1 X X 1 X d ku˛ k2˛  e˛ , u˛ /˛ . Aˇ uˇ , u˛ /˛ D .f (11.20) C A˛ 2 ˛D1 dt ˛D1 ˛D1 ˇ D1

For the second term in the left-hand side, we have  p1  p1 X  p1 X X  p1 X    Aˇ uˇ , u˛ /˛ D Aˇ uˇ , A˛ u˛ /p A˛ ˛D1

˛D1

ˇ D1

D

 p1 X

ˇ D1

Aˇ uˇ ,

ˇ D1

p1 X

A˛ u˛ /p

  0.

˛D1

For the right-hand side of (11.20), we apply the estimate  p1 1=2  p1 1=2 p1 p1 X X X X 2 2 e e e .f ˛ , u˛ /˛  kf ˛ k˛ ku˛ k˛  kf ˛ k˛ ku˛ k˛ . ˛D1

˛D1

˛D1

˛D1

297

Section 11.1 Splitting schemes for a system of equations

Applying Gronwall’s lemma and taking into account the initial condition (11.4), from (11.20), it follows that 1=2  p1 1=2 Z t  p1 1=2  p1 X X X 2 0 2 2 e ku˛ .t /k˛  kv˛ k˛ C kf ˛ . /k˛ d , (11.21) ˛D1

0

˛D1

˛D1

which implies the stability of the solution of the problem (11.4), (11.19) with respect to the initial data and the right-hand side. It is convenient to interpret the system of equations (11.19) as an evolutionary equation for the vector u D ¹u1 , u2 , : : : , up1 º: du e .t /, C Au D f dt

0 < t  T,

(11.22)

e2 , : : : , f ep1 º, and, for the elements of the operator matrix A, we e1 , f e D ¹f where f have the representation A D ¹A˛ˇ º,

A˛ˇ D A˛ Aˇ ,

˛, ˇ D 1, 2, : : : , p  1.

(11.23)

On the direct sum of spaces [59] H D H1 ˚ H2 ˚    ˚ Hp1 , we put .u, v/ D

p X

.u˛ , v˛ /˛ ,

kuk D 2

˛D1

p X

ku˛ k2˛ .

˛D1

In this case, A D A   0 in H and the estimation (11.21) may be rewritten in the following more compact form: Z t 0 e . /kd , ku.t /k  kv k C kf (11.24) 0

where D We also present a priori estimates for the solution of the Cauchy problem for the system of equations (11.1), (11.2) that complement the above estimates (11.18) and (11.21). Multiply equation (11.1) scalarly in H˛ by u˛ , then multiply equation (11.2) scalarly in Hp by up , and next add them. This yields v0

0 º. ¹v10 , v20 , : : : , vp1

p1 p1 X X 1d ku˛ k2˛ D .fp , up /p C .f˛ , u˛ /˛ . kup kp2 C 2 dt ˛D1 ˛D1

In view of

p X ˛D1

we have

.f˛ , u˛ /˛ 

p X ˛D1

ku˛ k2˛

p 1X C kf˛ k2˛ , 4 ˛D1

p 1X d 2 2 kf˛ k2˛ . kup kp  2kup kp C dt 2 ˛D1

298

Chapter 11 Equations with pairwise adjoint operators

This implies an a priori estimate for the p-th component of the solution of the problem (11.1)–(11.3): Z p X 1 t 2 0 2 exp.2.t   // kf˛ . /k2˛ . (11.25) ku˛ .t /k˛  exp.2t /kvp kp C 2 0 ˛D1 Evidently, a similar estimate can be obtained for the problem (11.3), (11.16) taking into account a specific structure of the right-hand side of equation (11.16). To obtain a priori estimates for the other components of the solution of the system (11.1), (11.2), we differentiate (11.1) with respect to time and multiply it scalarly in H˛ by u˛ . This gives     df˛ dup 1d 2 D ku˛ k˛ C A˛ , u˛ , u˛ , ˛ D 1, 2, : : : , p  1. (11.26) 2 dt dt dt ˛ ˛ Multiplying equation (11.2) scalarly in Hp by A˛ u˛ , for the second term in the lefthand side of (11.26), we obtain  p1    X dup   D Aˇ uˇ , A˛ u˛ C .fp , A˛ u˛ /p . , u˛ A˛ dt ˛ p ˇ D1

Adding equations (11.26), we have  p1  p1 p1 X X 1X d 2   Aˇ u ˇ , A˛ u ˛ ku˛ k˛ C 2 ˛D1 dt p ˛D1 D

p1 X

˛D1

ˇ D1

df˛ , u˛ dt



˛

 p1  X  fp , A˛ u˛ . ˛D1

p

In view of  p1  p1   p1 X X X 1 A˛ u˛  Aˇ uˇ , A˛ u˛ C kfp kp2 ,  fp , 4 p p ˛D1 ˛D1 ˇ D1

we arrive at the inequality 2 p1  1 X d f˛  2 2  C 1 kfp k2 .  ku˛ k˛  2ku˛ k˛ C p  dt 2 ˛D1 dt ˛ 2 ˛D1 p1 X

Hence, the desired a priori estimate has the form p1 X

ku˛ .t /k2˛

 exp.2t /

˛D1



p1 X

˛D1   p1 X

kv˛0 .t /k2˛

1 C 2

Z 0

t

exp.2.t   //

   f˛ . / 2   C 1 kfp . /k2 d . p  dt  2 ˛ ˛D1

(11.27)

299

Section 11.1 Splitting schemes for a system of equations

In contrast to (11.21), the a priori estimate (11.27) is more accurate in accounting for specific features of the problem. In particular, the right-hand sides of the system of equations (11.1), (11.2) are estimated using weaker norms. The above examinations result in the following statement. Theorem 11.1. The individual components of the solution of the Cauchy problem .11.1/–.11.3/ satisfy the a priori estimates .11.25/, .11.27/.

11.1.4 Schemes with weights To solve numerically the operator-differential problem (11.1)–(11.2), we use conventional schemes with weights. Define a uniform grid in time: !  D ! [ ¹T º D ¹t n D n ,

n D 0, 1, : : : , N0 ,

N0 D T º

and let y n D y.t n /, t n D n . To apply a two-level scheme, equations (11.1), (11.2) are approximated by difference equations y˛n C A˛ ypn D f˛n , ypnC1  ypn 



p1 X

n D 0, 1, : : : , N , A˛ y˛n, D fpn, ,

˛ D 1, 2, : : : , p  1, n D 0, 1, : : : , N0  1.

(11.28)

(11.29)

˛D1

We apply notation y n, D y nC1 C .1   /y n , where  is a numerical parameter (weight), which is usually within 0    1. For simplicity, we restrict ourselves to the case of the same weight for all terms of equation (11.2). In view of (11.3), we supplement (11.29) with the initial condition yp0 D vp0 .

(11.30)

Theorem 11.2. The operator-difference scheme .11.28/–.11.30/ is unconditionally stable under the condition   1=2, and the difference solution, for sufficiently small  , satisfies the levelwise estimates kypnC1 kp2  exp.16  /kypn kp2 C 

p X

kf˛n, k2˛ ,

˛D1 p1 X ˛D1

ky˛nC1 k2˛ kp2

 exp.16  /

p1 X ˛D1

ky˛n k2˛

C

(11.31)   f˛nC1  f˛n 2  C  kf n, k2 .  p p   

p1 X ˛D1

˛

(11.32)

300

Chapter 11 Equations with pairwise adjoint operators

Proof. From (11.28), we have y˛n, C A˛ ypn, D f˛n, ,

n D 0, 1, : : : , N0  1,

˛ D 1, 2, : : : , p  1.

(11.33)

Similarly to the proof of the estimate (11.25), multiply equation (11.33) scalarly in H˛ by y˛n, , then multiply equation (11.29) scalarly in Hp by ypn, , and next add them. This yields ! p1 p1 X X ypnC1  ypn n, n, 2 n, n, C ky˛ k˛ D .fp , yp /p C .f˛n, , y˛n, /˛ . (11.34) , yp  ˛D1 ˛D1 For the upper bound of the last term in the right-hand side, we use the inequality p1 X

.f˛n, , y˛n, /˛ 

˛D1

p1 X

ky˛n, k2˛ C

˛D1

p1 1X kf n, k2 . 4 ˛D1 ˛ ˛

Taking into account that

  1 1 .y nC1  y n / C .y nC1 C y n / y n, D   2 2

and   1=2, from (11.34), we get kypnC1 kp2



kypn kp2

C

2 .fpn, , ypn, /p

p1  X C kf n, k2 . 2 ˛D1 ˛ ˛

We have 1 2.fpn, , ypn, /p  kfpn, kp2 C 2kypn, kp2 , 2 kypn, kp2  2 .kypnC1 kp2 C kypn kp2 / with   1=2. Thus, we receive the inequality .1  4  /kypnC1 kp2  .1 C 4  /kypn kp2 C

p  X kf n, k2 . 2 ˛D1 ˛ ˛

(11.35)

Without loss of generality, we can assume that 8   1, and therefore 1 C 4  1  exp.16  /. 1  4   , 2 1  4  This allows us to obtain, from (11.35), the levelwise stability estimate (11.31). To prove (11.32), we employ the relations ypnC1  ypnC1 y˛nC1  y˛n f nC1  f˛n C A˛ D ˛ ,    n D 0, 1, : : : , N0  1, ˛ D 1, 2, : : : , p  1

(11.36)

301

Section 11.1 Splitting schemes for a system of equations

that follow from (11.28). Multiply (11.36) scalarly in H˛ by y˛n, . This yields   nC1     nC1 ypnC1  ypnC1 n, f˛  f˛n n, y˛  y˛n n, C A˛ D . , y˛ , y˛ , y˛    ˛ ˛ ˛ (11.37) For the second term in the left-hand side of (11.37), we apply the representation  p1    X ypnC1  ypnC1 n, D Aˇ yˇn, , A˛ y˛n, C .fpn, , A˛ y˛n, /p , , y˛ A˛  ˛ p ˇ D1

which can be obtained by multiplying (11.29) scalarly in Hp by A˛ y˛n, . Summation of (11.37) over ˛ D 1, 2, : : : , p  1 yields the equality  p1   p1 p1 X  y nC1  y n X X ˛ ˛ n,  n,  n, C Aˇ yˇ , A˛ y˛ , y˛  ˛ p ˛D1 ˛D1 ˇ D1 (11.38)     p1 p1 X f nC1  f n X ˛ ˛ D  fpn, , A˛ y˛n, . , y˛n,  ˛ p ˛D1 ˛D1 Taking into account that  p1    p1 p1 X X X 1 A˛ y˛n,  Aˇ yˇn, , A˛ y˛n, C kfpn, kp2 , fpn, , 4 p p ˛D1 ˛D1 ˇ D1

for   1=2, from (11.38), we obtain p1 X

ky˛nC1 k2˛

˛D1



p1 X

ky˛n k2˛

C 2

˛D1

p1 X ˛D1

f˛nC1  f˛n n, , y˛ 



 C kfpn, kp2 . 2 ˛

Similarly to (11.35), we establish the inequality .1  4  /

p1 X ˛D1

ky˛nC1 k2˛ kp2

 .1 C 4  /

p1 X

ky˛n k2˛

˛D1 p1 X  f˛nC1 

 C 2 ˛D1 

2  f˛n   C  kf n, k2 .   2 p p ˛

For small  (8   1), from this estimate, it follows that the desired estimate (11.32) holds. The proven estimates for stability of the individual components of the difference solution for (11.28)–(11.30) may be treated as discrete analogs of the estimates (11.25), (11.27) for the solution of the Cauchy problem for the system of equations (11.1), (11.2).

302

Chapter 11 Equations with pairwise adjoint operators

The simplest computational implementation of the operator-difference scheme (11.28)–(11.30) may be based on elimination of the individual components of the solution. Direct substitution of y˛n , ˛ D 1, 2, : : : , p  1 from (11.33) into (11.29) yields a discrete analog of equation (11.16): ypnC1  ypn 

C

p1 X

A˛ A˛ ypn, D fQpn, ,

(11.39)

˛D1

under the condition that en, D f n, C f p p

p1 X

A˛ f˛n, ,

n D 0, 1, : : : , N  1.

˛D1

To find ypnC1 from (11.39), we solve the grid problem ypnC1 C  

p1 X

A˛ A˛ ypnC1 D 'pn .

(11.40)

˛D1

After defining ypnC1 , other components of the solution y˛nC1 , ˛ D 1, 2, : : : , p  1 are evaluated by (11.28). The second principal possibility of the numerical implementation of the operatordifference scheme (11.28)–(11.30) is associated with the elimination of ypnC1 , i.e., with the transition to the discrete analog of the system (11.19). Multiplying (11.29) by A˛ and using (11.36), we obtain the system of equations p1 X y˛nC1  y˛n en, , A˛ y˛n, D f C A˛ ˛  ˛D1

(11.41)

where nC1  f n ˛ en, D f˛ f  A˛ fpn, , ˛ 

˛ D 1, 2, : : : , p  1.

To determine y˛nC1 , ˛ D 1, 2, : : : , p 1, from (11.41), we obtain the discrete problem y˛nC1

C  A˛

p1 X ˇ D1

Aˇ yˇnC1 D '˛n ,

˛ D 1, 2, : : : , p  1.

(11.42)

Further, from (11.29), we evaluate ypnC1 . The computational complexity of the problem (11.42) is significantly higher in comparison with (11.40). In the case (11.42), we must solve a system of strongly coupled equations instead of the single equation (11.40). Therefore, the implementation of the scheme with weights (11.28)–(11.30) should be based on (11.39).

303

Section 11.1 Splitting schemes for a system of equations

11.1.5 Splitting schemes to find the p-th component of the solution Additive difference schemes are constructed to reduce the computational costs during the transition to a new time level in the numerical solving of unsteady problems. Considering the problem (11.1)–(11.3), it is natural to design splitting schemes that are based on solving a sequence of simpler problems associated with the individual operators A˛ , A˛ , ˛ D 1, 2, : : : , p  1. For the standard schemes with weights (11.28)– (11.30), the corresponding problems at the new time level (see (11.40) or (11.42)) are not based on these individual operators; they employ their combinations. The construction of additive schemes for the problem (11.1)–(11.3) is conducted considering problems for the individual components. It is convenient to start with the problem (11.3), (11.16). We write equation (11.16) as X dup ep , B˛ up D f C dt ˛D1 p1

0 < t  T,

(11.43)

where each operator term B˛ , ˛ D 1, 2, : : : , p  1 is a self-adjoint and non-negative operator acting from Hp onto Hp : B˛ D A˛ A˛  0,

˛ D 1, 2, : : : , p  1.

(11.44)

Approaches to the construction of splitting schemes for the problem (11.3), (11.43), (11.44) are well developed. As usual, we separate the case of two-component (p D 3) splitting in (11.43). In this situation, we can orient ourself to the operator analogs of the classical schemes of alternating direction [28, 104]. For numerically solving the problem (11.3), (11.43) with p D 3, we can use the factorized scheme .Ep C  B1 /.Ep C  B2 /

y nC1  y n en, , C .B1 C B2 /y n D f p 

(11.45)

where Ep is the identity operator in Hp . The main result on the stability of the factorized scheme is formulated as follows. Theorem 11.3. The factorized operator-difference scheme .11.44/, .11.45/ for solving the problem .11.3/, .11.16/ with p D 3 is unconditionally stable for   1=2, and the difference solution satisfies the levelwise estimates en, k2 . (11.46) k.Ep C  B2 /y nC1 kp2  k.Ep C  B2 /y n kp2 C  k.Ep C  B1 /1 f p p On the basis of the a priori estimate (11.46), we establish the convergence of the difference solution to the exact one for the problem (11.3), (11.16) with accuracy O. 2 / for  D 1=2, and with accuracy O. / in the case of  ¤ 1=2. Additive difference schemes for problems with splitting into three and more pairwise noncommutative operators are based on the full approximation concept, i.e., they

304

Chapter 11 Equations with pairwise adjoint operators

are of componentwise splitting (locally one-dimensional schemes). For the problem (11.43),(11.44), we employ the scheme y nC˛=.p1/  y nC.˛1/=.p1/ C B˛ .˛ y nC˛=.p1/ C .1  ˛ /y nC.˛1/=.p1/ /  D '˛n , ˛ D 1, 2, : : : , p  1, n D 0, 1, : : : , N  1, (11.47) where the right-hand side is en, D f p

p1 X

'˛n .

˛D1

If ˛  1=2, then the componentwise splitting scheme (11.47) is unconditionally stable. Here is the relevant a priori estimate for stability with respect to the initial data and the right-hand side. For the right-hand sides '˛n , ˛ D 1, 2, : : : , p  1, we use the special representation '˛n D 'N˛n C 'Q˛n ,

p1 X

˛ D 1, 2, : : : , p  1,

'Q˛n D 0.

(11.48)

˛D1

This form of the right-hand side is of fundamental importance in considering the problem for the error of the additive scheme. For the scheme of componentwise splitting, the following statement is true. Theorem 11.4. If 2  ˛  1=2, ˛ D 1, 2, : : : , p  1, then the difference solution of the problem .11.3/, .11.47/, .11.48/ satisfies the a priori estimate ky

nC1

n

kp  ky kp C 

p1 X ˛D1

k'N˛k kp

 p1  X   k C  B˛ 'Qˇ  . ˇ D˛

p

(11.49)

If we focus on using parallel computers, then we should employ additively averaged schemes of componentwise splitting. In this case, the transition to a new time level is performed as follows: y˛nC1  y n C B˛ .˛ y˛nC1 C .1  ˛ /y n / D '˛n , .p  1/ ˛ D 1, 2, : : : , p  1, n D 0, 1, : : : , N  1, and y nC1 D

p1 X 1 y nC1 . .p  1/ ˛D1 ˛

(11.50)

(11.51)

The stability conditions for these schemes are the same as for the standard schemes of componentwise splitting. Similarly to Theorem 11.4, we prove the following statement.

305

Section 11.1 Splitting schemes for a system of equations

Theorem 11.5. If ˛  1=2, ˛ D 1, 2, : : : , p  1, then the solution of the problem .11.3/, .11.48/, .11.50/, .11.51/ satisfies the a priori estimate ky

nC1

n

kp  ky kp C 

     k'N˛k kp C p ˛ B˛ 'Q˛k  .

p1 X ˛D1

p

(11.52)

A potential advantage of additively averaged schemes (11.50), (11.51) results from the fact that we can arrange parallel computations for evaluating grid functions y˛nC1 , ˛ D 1, 2, : : : , p  1. The above estimates for stability (11.49) and (11.52) provide the basis for studying accuracy of the splitting schemes under discussion. We formulate the problem for the error of the difference solution in the form (11.47) and (11.50), (11.51). The fundamental point is that, in the above componentwise splitting schemes and additively averaged schemes, the estimates for stability depend essentially on the splitting (11.48), i.e., they depend on the form of the individual terms. In fact, this means that accuracy of these additive schemes depends on the way that is used to formulate the intermediate problems, it depends on approximations to these intermediate problems, and so on. On the other hand, the intermediate problems (the auxiliary grid values y nC˛=.p1/ , y˛n , ˛ D 1, 2, : : : , p  1) have no independent sense. Ideally, we should try to construct schemes without them, i.e., without involving the full approximation concept. Some possibilities in this direction are associated with the regularized additive schemes, which are based on the regularization principle for operator-difference schemes. Now we construct additive schemes via a perturbation of each individual operator term in the additive representation (11.43): y nC1  y n X en, , .Ep C ˛ B˛ /1 B˛ y n D f C p  ˛D1 p1

n D 0, 1, : : : , N  1. (11.53)

Theorem 11.6. If ˛  .p  1/=2, ˛ D 1, 2, : : : , p  1, then the additive scheme .11.53/ is unconditionally stable, and the difference solution satisfies the a priori estimate en, kp . (11.54) ky nC1 kp  ky n kp C  kf p For the regularized additive scheme (11.53), the estimate for stability with respect to the initial data and the right-hand side (11.54) has no relation with the splitting of the right-hand side (11.48). The factorized scheme (11.45) demonstrates definite advantages over the additive multicomponent splitting schemes (11.47), (11.50) and (11.53) above, which have the first-order approximation in time. The first preference is attributed to the fact that there is no problem to construct an algorithm with the second-order approximation – it is

306

Chapter 11 Equations with pairwise adjoint operators

sufficient to put  D 1=2. For the schemes of type (11.47), this is not enough; we must also arrange a special scheme of calculations: B1 ! B2 !    ! Bp1 ! Bp1 !    ! B2 ! B1 . The construction of additively averaged operator schemes (see (11.50)) with the second-order approximation in time seems to be very difficult. Regularized additive schemes (see (11.53)) of second order are designed using three-level operatordifference schemes. The second and more important benefit of factorized schemes results from the fact that they are schemes based on a pseudo-time evolution process. For problems with a constant right-hand side, the solution of the unsteady problem tends to the solution of the steady-state problem as t ! 1. This is true for the discrete solution (after discretization in time). Among the additive operator-difference schemes with the general multicomponent splitting, vector additive schemes belong to the class of schemes based on a pseudo-time evolution process. However, their construction leads to significant complication of the original problem – the transition from a single equation to a system of equations increases drastically the computational costs.

11.1.6 Additive schemes for systems of equations Let us discuss possibilities of constructing operator-splitting schemes for the problem (11.1)–(11.3) based on the elimination of the component up . This would solve the problem (11.4), (11.19). It is convenient to handle the system of equations (11.19) in vector form (11.22), (11.23). For the self-adjoint matrix operator A, we use the triangular decomposition A1 D A2 .

A D A1 C A2 ,

(11.55)

By (11.23), we have 0

1  2 A1 A1  A2 A1

B A1 D B @  Ap1 A1 0

1 A1 A1 B2

B 0 A2 D B B  @ 0

0 1  A 2 2 A2  Ap1 A2 A1 A2 1  2 A2 A2  0

   

   

0 0 0

1 C C, A

1  2 Ap1 Ap1

1  A1 Ap1 C  C A2 Ap1 C. C  Ap2 Ap2 A 1  A A 2 p1 p1

Section 11.1 Splitting schemes for a system of equations

307

For numerically solving the Cauchy problem for equation (11.22), we apply the factorized operator-difference scheme (the alternating triangle method), i.e., .E C   A1 /.E C   A2 /

y nC1  y n e n, , C Ay n D f 

y 0 D v0 ,

(11.56) (11.57)

where E is the identity operator in H . If  D 1=2, then the difference equation (11.56) approximates the operator-differential equation (11.22) with the second order by  , whereas in the opposite case  ¤ 1=2, we have the first order. Theorem 11.7. The factorized operator-difference scheme .11.55/–.11.57/ is unconditionally stable for   1=2, and the difference solution satisfies the following estimate for stability with respect to the initial data and the right-hand side: e n, k, n D 0, 1, : : : , N  1, ky nC1 kB  ky n kB C  kf

(11.58)

where  B D .E C   A2 /.E C   A2 /  .A1 C A2 /, 2  B D B  E. Proof. In view of the above notation, rewrite (11.56) as B

y nC1  y n y nC1 C y n e n, . CA Df  2

Multiplying this equation scalarly in H by  .y nC1 C y n / and taking into account the non-negativity of the operator A, we obtain e n, , .y nC1 C y n //. ky nC1 k2B  ky n k2B   .f Taking into account that .fQ n, , .y nC1 C y n //  kfQ n, k.ky nC1 k C ky n k/, ky nC1 k C ky n k  ky nC1 kB C ky n kB , ky nC1 k2B  ky n k2B D ky nC1 kB  ky n kB ky nC1 kB C ky n kB , we establish the desired estimate (11.58). The estimate (11.58) is just a discrete analog of the a priori estimate (11.24) for the solution of the Cauchy problem formulated for the operator-differential equation (11.22). The computational implementation of the factorized scheme (11.56) involves

308

Chapter 11 Equations with pairwise adjoint operators

the inversion of the operators E C   A1 and E C   A2 . In view of the triangular structure of the operators A1 , A2 , this leads to the successive solution of the problems y˛nC1=2 C  A˛ A˛ y˛nC1=2 D '˛n , yˇnC1 C  Aˇ Aˇ yˇnC1 D

n ˇ,

˛ D 1, 2, : : : , p  1,

ˇ D p  1, p  2, : : : , 1.

The above-considered additive schemes of the second-order approximation in time for finding the single component up are implemented in a similar way. Thus, the computational complexity of solving the Cauchy problem for the system of equations (11.19) (the components u˛ , ˛ D 1, 2, : : : , p 1 are determined) is comparable to the computational complexity of solving the problem (11.3), (11.16) for the single component. It is important to note also that the resulting factorized operator-difference scheme (11.56), (11.57) belongs to the class of schemes that are based on a pseudo-time evolution process. Among other additive schemes for solving the problem (11.4), (11.19), we highlight the scheme  y˛n X en, , A˛ Aˇ yˇn D f C ˛  p1

y nC1 .Ep C A˛ A˛ / ˛

˛ D 1, 2, : : : , p1. (11.59)

ˇ D1

In this case, only the diagonal part of the operator A is referred to a new time level. Using vector notation, the scheme (11.59) may be written as .E C   D/ where D D ¹D˛ˇ º,

y nC1  y n e n, , C Ay n D f 

D˛ˇ D A˛ Aˇ ı˛ˇ ,

(11.60)

˛, ˇ D 1, 2, : : : , p  1,

and ı˛ˇ is Kronecker’s symbol. Theorem 11.8. The factorized operator-difference scheme .11.57/, .11.60/ is unconditionally stable if   .p  1/=2, and the difference solution satisfies the estimate for stability with respect to the initial data and the right-hand side .11.58/, where  B D E C   D  A, B D B   E . 2 Proof. The proof is similar to Theorem 11.7. It is sufficient to show that if   .p  1/=2, then we have B  E . In doing so, we involve the inequality  p1  p1 X X   A˛ y˛ , A˛ y˛ .Ay, y/ D ˛D1

 .p  1/

˛D1 p1 X ˛D1

p

.A˛ y˛ , A˛ y˛ /p D .p  1/.Dy, y/.

309

Section 11.1 Splitting schemes for a system of equations

The main potential advantage of the above-considered additive scheme (11.57), (11.60) consists in asynchronous computations of the components of the approximate solution at the new time level – it is very easy to arrange parallel computations. This scheme has the first-order approximation in time. The additive scheme of the secondorder approximation from the class of three-level schemes is constructed and investigated in a similar manner. In view of the above notation, we put y nC1  2y n C y n1 y nC1  y n1 e n. C Ay n D f (11.61) C   2D 2 2 The three-level operator-difference scheme (11.61) with the self-adjoint and positive operators A and D is stable in the appropriate spaces if the following inequality holds: 1  D > A. 4 In view of the above-established inequality A  .p  1/D, it is sufficient to choose  > .p  1/=4. Now we analyze some additive schemes for the original problem (11.1)–(11.3) associating them with the above-constructed schemes of splitting with respect to the individual components of the solution. For example, the componentwise splitting scheme (11.47) for the single equation (11.43), (11.44) may be attributed to the following additive scheme for the system of equations (11.1), (11.2): .Ep C  A˛ A˛ /

nC˛=.p1/

yp

nC.˛1/=.p1/

 yp 

 A˛ y˛nC.˛1/=.p1/ D '˛n ,

y˛nC.˛1/=.p1/ C A˛ ypnC.˛1/=.p1/ D f˛n, , ˛ D 1, 2, : : : , p  1,

n D 0, 1, : : : , N  1.

Analogs of the other additive schemes for finding the approximate solution of the problem (11.3), (11.16) are constructed in a similar way. For the additive scheme (11.59) to solve the problem (11.4), (11.19), we put into the correspondence the scheme .E˛ C  A˛ A˛ /y˛nC1 C A˛ ypn D f˛n, , ypnC1  ypn 



p1 X

˛ D 1, 2, : : : , p  1.

A˛ y˛n D fpn,

˛D1

for numerically solving the problem (11.1)–(11.3). A slightly more complicated construction is used as an analog of the factored scheme nC1=2 and put (11.56), (11.57). We introduce the auxiliary quantities y˛   nC1=2 ˛1 X ypnC1  ypn 1 f nC1  f˛n  y˛n y˛ Aˇ Aˇ C  A˛ A˛ . C A˛ D ˛ E˛ C   2    ˇ D1

310

Chapter 11 Equations with pairwise adjoint operators

To approximate (11.2), we apply ypnC1  ypn 



p1 X

A˛ y˛n D fpn, .

˛ D 1, 2, : : : , p  1.

˛D1

The solution at the new time level is defined according to  nC1  p1 nC1=2 nC1=2 X 1 y˛  y˛  y˛n   y˛ Aˇ Aˇ D , E˛ C  A˛ A˛ C   2   ˇ D˛C1

˛ D p  1, p  2, : : : , 1. It should be noted that we have discussed only the problem of algebraic equivalence between the above-constructed additive schemes for the individual components of the solution and some versions of additive schemes for the original system of equations (11.1), (11.2). The issues of approximation and convergence require a more detailed consideration. The main result of our study is that the additive schemes constructed for the system of equations (for finding the components u˛ , ˛ D 1, 2, : : : , p  1) seem to be preferable in comparison with the additive schemes developed for the single equation (for the component up ).

11.2 Additive schemes for a system of first-order equations We consider difference approximations in time for numerical solving of the Cauchy problem associated with a special system of evolutionary equations of first order. Such problems result from the approximation in space of the Schroedinger equation for unsteady problems of acoustics and electrodynamics where the imaginary and real parts are separated. First, we construct for these equations unconditionally stable two-level operator-difference schemes with weights. The second class of difference schemes under investigation is based on the formal transition to explicit operator-difference schemes for an evolutionary equation of second order with explicit-implicit approximations for individual equations of the system. The regularization of these schemes is applied in order to obtain unconditionally stable operatordifference schemes. Operator-splitting schemes are constructed that are based on the solution of elementary problems at every time step.

11.2.1 Statement of the problem Below, we study (see also [200]) the Cauchy problem for a system of first-order linear equations with a skew-symmetric problem operator in a Euclidean space. A specific feature of the problem consists in a special structure of the operator matrix with nonzero elements in the last column and in the last row. Here we present elementary a

311

Section 11.2 Additive schemes for a system of first-order equations

priori estimates for the solution, which serve as a guide in constructing a discrete problem. The operator form under discussion is applicable to typical problems of acoustics and electrodynamics. We consider the classical schemes with weights with emphasis on their computational implementation. Explicit-implicit schemes are investigated separately on distinct time grids: grids with integer nodes and meshes with half-integer grid points. Two classes of additive schemes are proposed to design techniques based on solving problems related to individual operators. The first class of schemes is associated with the standard additive splitting of the operator matrix of the problem. The second class of additive schemes is based on the transition to an equivalent evolutionary equation of second order. Let H˛ , ˛ D 1, 2, : : : , p be finite-dimensional real Hilbert (Euclidean) spaces, where the scalar product and norm are ., /˛ and k  k˛ , ˛ D 1, 2, : : : , p, respectively. The individual components of the solution are denoted by u˛ .t /, ˛ D 1, 2, : : : , p for any t (0  t  T , T > 0). We seek the solution of the system of evolutionary first-order equations du˛ C A˛ up D f˛ , ˛ D 1, 2, : : : , p  1, dt p1 dup X  A˛ u˛ D fp , 0 < t  T .  dt ˛D1

(11.62)

Here f˛ .t / 2 L2 .0, T ; H˛ /, ˛ D 1, 2, : : : , p are given, and A˛ is a linear constant (independent of t ) operator acting from Hp onto H˛ for each ˛ D 1, 2, : : : , p  1. The system of equations (11.62) is supplemented by initial conditions u˛ .0/ D v˛0 ,

˛ D 1, 2, : : : , p.

(11.63)

Let us present elementary a priori estimates for the solution of the Cauchy problem (11.62), (11.63), which provide a checkpoint in studying operator-difference schemes. Multiplying all individual equations in (11.62) scalarly in H˛ by u˛ and adding them, we obtain p p X 1 X d ku˛ k2˛ .f˛ , u˛ /˛ . D 2 ˛D1 dt ˛D1 For the right-hand side, we use the estimate X 1=2  X 1=2 p p p p X X 2 2 .f˛ , u˛ /˛  kf˛ k˛ ku˛ k˛  kf˛ k˛ ku˛ k˛ . ˛D1

˛D1

˛D1

˛D1

Applying Gronwall’s lemma and taking into account the initial condition (11.63), we obtain the estimate Z tX p p p X X 2 0 2 ku˛ .t /k˛  kv˛ k˛ C kf˛ . /k2˛ d , (11.64) ˛D1

˛D1

0 ˛D1

312

Chapter 11 Equations with pairwise adjoint operators

which implies the stability of the solution of the problem (11.62), (11.63) with respect to the initial data and the right-hand side. In some cases, it is convenient to interpret the system of equations (11.62) as a single evolutionary equation for the vector u D ¹u1 , u2 , : : : , up º: du C Au D f .t /, dt

0 < t  T,

(11.65)

where f D ¹f1 , f2 , : : : , fp º, and, for the elements of the operator matrix A, we have the representation A D ¹A˛ˇ º, 8 0, ˆ ˆ