Vabishchevich Additive Operator-Difference Schemes

Splitting Schemes
1. Auflage 2013
ISBN: 978-3-11-032146-3
Verlag: De Gruyter
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 370 Seiten

Applied mathematical modeling is concerned with solving unsteady problems. Splitting schemes are attributed to the transition from a complex problem to a chain of simpler problems. This book shows how to construct additive difference schemes (splitting schemes) to solve approximately 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. Also regionally additive schemes (domain decomposition methods) and unconditionally stable additive schemes of multi-component splitting are considered for evolutionary equations of first and second order as well as for systems of equations.

The book is written for specialists in computational mathematics and mathematical modeling. All topics are presented in a clear and accessible manner.

Zielgruppe

Researchers in computational mathematics and mathematical modeling; academic libraries.

Autoren/Hrsg.

Vabishchevich, Petr N.

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen

Inhaltsverzeichnis

1;Preface;5
2;Notation;15
3;1 Introduction;17
3.1;1.1 Numerical methods;17
3.2;1.2 Additive operator-difference schemes;19
3.3;1.3 The main results;22
3.4;1.4 Contents of the book;26
4;2 Stability of operator-difference schemes;30
4.1;2.1 The Cauchy problem for an operator-differential equation;30
4.1.1;2.1.1 Hilbert spaces;30
4.1.2;2.1.2 Linear operators in a finite-dimensional space;32
4.1.3;2.1.3 Operators in a finite-dimensional Hilbert space;33
4.1.4;2.1.4 The Cauchy problem for an evolutionary equation of first order;35
4.1.5;2.1.5 Systems of linear ordinary differential equations;36
4.1.6;2.1.6 A boundary value problem for a one-dimensional parabolic equation;37
4.1.7;2.1.7 Equations of second order;39
4.2;2.2 Two-level schemes;40
4.2.1;2.2.1 Key concepts;40
4.2.2;2.2.2 Stability with respect to the initial data;42
4.2.3;2.2.3 Stability with respect to the right-hand side;45
4.2.4;2.2.4 Schemes with weights;47
4.3;2.3 Three-level schemes;48
4.3.1;2.3.1 Stability with respect to the initial data;48
4.3.2;2.3.2 Reduction to a two-level scheme;50
4.3.3;2.3.3 P-stability of three-level schemes;52
4.3.4;2.3.4 Estimates in simpler norms;54
4.3.5;2.3.5 Stability with respect to the right-hand side;56
4.3.6;2.3.6 Schemes with weights for equations of first order;56
4.3.7;2.3.7 Schemes with weights for equations of second order;58
4.4;2.4 Stability in finite-dimensional Banach spaces;59
4.4.1;2.4.1 The Cauchy problem for a system of ordinary differential equations;59
4.4.2;2.4.2 Scheme with weights;61
4.4.3;2.4.3 Difference schemes for a one-dimensional parabolic equation;63
4.5;2.5 Stability of projection-difference schemes;63
4.5.1;2.5.1 Preliminary observations;64
4.5.2;2.5.2 Stability of finite element techniques;65
4.5.3;2.5.3 Stability of projection-difference schemes;67
4.5.4;2.5.4 Conditions for -stability of projection-difference schemes;69
4.5.5;2.5.5 Schemes with weights;71
4.5.6;2.5.6 Stability with respect to the right-hand side;73
4.5.7;2.5.7 Stability of three-level schemes with respect to the initial data;75
4.5.8;2.5.8 Stability with respect to the right-hand side;76
4.5.9;2.5.9 Schemes for an equation of first order;77
5;3 Operator splitting;79
5.1;3.1 Time-dependent problems of convection-diffusion;79
5.1.1;3.1.1 Differential problem;79
5.1.2;3.1.2 Semi-discrete problem;84
5.1.3;3.1.3 Two-level schemes;86
5.2;3.2 Splitting operators in convection-diffusion problems;93
5.2.1;3.2.1 Splitting with respect to spatial variables;93
5.2.2;3.2.2 Splitting with respect to physical processes;94
5.2.3;3.2.3 Schemes for problems with an operator semibounded from below;96
5.3;3.3 Domain decomposition methods;98
5.3.1;3.3.1 Preliminaries;98
5.3.2;3.3.2 Model boundary value problems;101
5.3.3;3.3.3 Standard finite difference approximations;103
5.3.4;3.3.4 Domain decomposition;107
5.3.5;3.3.5 Problems with non-self-adjoint operators;114
5.4;3.4 Difference schemes for time-dependent vector problems;117
5.4.1;3.4.1 Preliminary discussions;117
5.4.2;3.4.2 Statement of the problem;118
5.4.3;3.4.3 Estimates for the solution of differential problems;120
5.4.4;3.4.4 Approximation in space;122
5.4.5;3.4.5 Schemes with weights;124
5.4.6;3.4.6 Alternating triangle method;125
5.5;3.5 Problems of hydrodynamics of an incompressible fluid;128
5.5.1;3.5.1 Differential problem;128
5.5.2;3.5.2 Discretization in space;130
5.5.3;3.5.3 Peculiarities of hydrodynamic equations written in the primitive variables;133
5.5.4;3.5.4 A priori estimate for the differential problem;134
5.5.5;3.5.5 Approximation in space;135
5.5.6;3.5.6 Additive difference schemes;137
6;4 Additive schemes of two-component splitting;139
6.1;4.1 Alternating direction implicit schemes;139
6.1.1;4.1.1 Problem formulation;139
6.1.2;4.1.2 The Peaceman–Rachford scheme;140
6.1.3;4.1.3 Stability of the Peaceman–Rachford scheme;141
6.1.4;4.1.4 Accuracy of the Peaceman–Rachford scheme;142
6.1.5;4.1.5 Another ADI scheme;143
6.2;4.2 Factorized schemes;143
6.2.1;4.2.1 General considerations;144
6.2.2;4.2.2 ADI methods as factorized schemes;144
6.2.3;4.2.3 Stability and accuracy of factorized schemes;145
6.2.4;4.2.4 Regularization principle for constructing factorized schemes;147
6.2.5;4.2.5 Factorized schemes of multicomponent splitting;149
6.3;4.3 Alternating triangle method;150
6.3.1;4.3.1 General description of the alternating triangle method;151
6.3.2;4.3.2 Investigation of stability and convergence;152
6.3.3;4.3.3 Three-level additive schemes;153
6.3.4;4.3.4 Problems with non-self-adjoint operators;155
6.4;4.4 Equations of second order;156
6.4.1;4.4.1 Model problem;157
6.4.2;4.4.2 Factorized schemes;158
6.4.3;4.4.3 Schemes of the alternating triangle method;159
7;5 Schemes of summarized approximation;160
7.1;5.1 Additive formulations of differential problems;160
7.1.1;5.1.1 Model problem;160
7.1.2;5.1.2 Intermediate problems;161
7.1.3;5.1.3 Summarized approximation concept;163
7.1.4;5.1.4 Schemes of the second-order summarized approximation;164
7.2;5.2 Investigation of schemes of summarized approximation;166
7.2.1;5.2.1 Schemes of componentwise splitting;166
7.2.2;5.2.2 Estimates for the intermediate problem solutions;167
7.2.3;5.2.3 Stability of componentwise splitting schemes;169
7.2.4;5.2.4 Convergence of componentwise splitting schemes;170
7.2.5;5.2.5 Convergence of additive schemes in Banach spaces;171
7.3;5.3 Additively averaged schemes;172
7.3.1;5.3.1 Differential problem;172
7.3.2;5.3.2 Additive schemes;173
7.3.3;5.3.3 Stability of additively averaged schemes;174
7.4;5.4 Other variants of componentwise splitting schemes;176
7.4.1;5.4.1 Fully implicit additive schemes;176
7.4.2;5.4.2 ADI methods as additive schemes;177
7.4.3;5.4.3 Additive schemes with second-order accuracy;178
7.4.4;5.4.4 Convergence of higher-order schemes;179
8;6 Regularized additive schemes;183
8.1;6.1 Multiplicative regularization of difference schemes;183
8.1.1;6.1.1 Regularization principle for difference schemes;183
8.1.2;6.1.2 Additive regularization;184
8.1.3;6.1.3 Multiplicative regularization;186
8.2;6.2 Multiplicative regularization of additive schemes;187
8.2.1;6.2.1 The Cauchy problem for a first-order equation;187
8.2.2;6.2.2 Regularization of additive schemes;188
8.2.3;6.2.3 Stability and convergence;189
8.2.4;6.2.4 Regularized and additively averaged schemes;191
8.3;6.3 Schemes of higher-order accuracy;192
8.3.1;6.3.1 Statement of the problem;192
8.3.2;6.3.2 Explicit three-level scheme;193
8.3.3;6.3.3 Regularized schemes;194
8.3.4;6.3.4 Additively averaged scheme;195
8.4;6.4 Regularized schemes for equations of second order;196
8.4.1;6.4.1 Model problem;196
8.4.2;6.4.2 Regularized scheme;197
8.4.3;6.4.3 Additively averaged schemes for equations of second order;198
8.5;6.5 Regularized schemes with general regularizers;199
8.5.1;6.5.1 General regularizers;199
8.5.2;6.5.2 Additive schemes with a general-form regularizer;201
8.5.3;6.5.3 Factorized additive schemes;202
8.5.4;6.5.4 Generalizations;203
9;7 Schemes based on approximations of a transition operator;206
9.1;7.1 Operator-difference schemes;206
9.1.1;7.1.1 Operator-differential problem;206
9.1.2;7.1.2 Difference approximations in time;207
9.1.3;7.1.3 SM-stable schemes for problems with a self-adjoint operator;210
9.1.4;7.1.4 Factorized SM-stable two-level schemes;215
9.1.5;7.1.5 Problems with a skew-symmetric operator;219
9.2;7.2 Additive schemes with a multiplicative transition operator;220
9.2.1;7.2.1 Operator-differential problems;220
9.2.2;7.2.2 Componentwise splitting schemes;222
9.3;7.3 Splitting schemes with an additive transition operator;224
9.3.1;7.3.1 Additive approximation of a transition operator;225
9.3.2;7.3.2 Additive schemes;225
9.3.3;7.3.3 Regularized additive schemes;227
9.4;7.4 Further additive schemes;227
9.4.1;7.4.1 Schemes of the second order;228
9.4.2;7.4.2 Factorized schemes;229
9.4.3;7.4.3 Inhomogeneous approximation of a transition operator;230
9.4.4;7.4.4 Schemes of higher-order approximation;231
10;8 Vector additive schemes;234
10.1;8.1 Vector schemes for first-order equations;234
10.1.1;8.1.1 Vector differential problem;234
10.1.2;8.1.2 Stability of vector additive schemes;236
10.1.3;8.1.3 Stability with respect to the right-hand side;239
10.2;8.2 Stability of vector additive schemes in Banach spaces;240
10.2.1;8.2.1 Problem formulation;240
10.2.2;8.2.2 Vector additive scheme;241
10.2.3;8.2.3 Study on stability;242
10.3;8.3 Schemes of second-order accuracy;244
10.3.1;8.3.1 Statement of the problem;244
10.3.2;8.3.2 Three-level vector schemes;245
10.3.3;8.3.3 Schemes of the alternating triangle method;247
10.4;8.4 Vector schemes for equations of second order;248
10.4.1;8.4.1 The Cauchy problem for a second-order equation;248
10.4.2;8.4.2 Vector problem;250
10.4.3;8.4.3 Scheme with weights;251
10.4.4;8.4.4 Additive schemes;252
10.4.5;8.4.5 Stability of additive schemes;254
11;9 Iterative methods;256
11.1;9.1 Basics of iterative methods;256
11.1.1;9.1.1 Problem formulation;256
11.1.2;9.1.2 Simple iteration method;258
11.1.3;9.1.3 The Chebyshev iterative method;259
11.1.4;9.1.4 Two-level variation-type methods;260
11.1.5;9.1.5 Conjugate gradient method;261
11.2;9.2 Iterative alternating direction method;262
11.2.1;9.2.1 Iterative method with two-component splitting;262
11.2.2;9.2.2 Convergence study;263
11.2.3;9.2.3 Modified iterative method of alternating directions;265
11.2.4;9.2.4 Multicomponent splitting;266
11.3;9.3 Iterative alternating triangle method;268
11.3.1;9.3.1 Iterative method;268
11.3.2;9.3.2 Convergence rate;269
11.3.3;9.3.3 Modified iterative method of alternating triangles;271
11.4;9.4 Iterative cluster aggregation methods;271
11.4.1;9.4.1 Transition to a system of equations;272
11.4.2;9.4.2 Iterative method;273
11.4.3;9.4.3 Parallel variant;275
11.4.4;9.4.4 Aggregation of unknowns;276
12;10 Splitting of the operator at the time derivative;279
12.1;10.1 Schemes with splitting of the operator at the time derivative;279
12.1.1;10.1.1 Preliminary discussions;279
12.1.2;10.1.2 Statement of the problem;280
12.1.3;10.1.3 Vector problem;282
12.1.4;10.1.4 Vector additive schemes;284
12.1.5;10.1.5 Generalizations;288
12.2;10.2 General splitting;288
12.2.1;10.2.1 Preliminary discussions;289
12.2.2;10.2.2 Problem formulation;290
12.2.3;10.2.3 Scheme with weights;292
12.2.4;10.2.4 Schemes with a diagonal operator;294
12.2.5;10.2.5 The more general problem;295
12.3;10.3 Explicit-implicit splitting schemes;298
12.3.1;10.3.1 Introduction;298
12.3.2;10.3.2 Boundary value problems for systems of equations;299
12.3.3;10.3.3 Schemes with a diagonal operator;301
12.3.4;10.3.4 General case;305
13;11 Equations with pairwise adjoint operators;307
13.1;11.1 Splitting schemes for a system of equations;307
13.1.1;11.1.1 Preliminary discussions;308
13.1.2;11.1.2 Statement of the problem;309
13.1.3;11.1.3 A priori estimates;311
13.1.4;11.1.4 Schemes with weights;315
13.1.5;11.1.5 Splitting schemes to find the p-th component of the solution;319
13.1.6;11.1.6 Additive schemes for systems of equations;322
13.2;11.2 Additive schemes for a system of first-order equations;326
13.2.1;11.2.1 Statement of the problem;326
13.2.2;11.2.2 Examples;329
13.2.3;11.2.3 Schemes with weights;332
13.2.4;11.2.4 Explicit-implicit schemes;334
13.2.5;11.2.5 Additive schemes of componentwise splitting;338
13.2.6;11.2.6 Regularized additive schemes;340
13.3;11.3 Another class of systems of first-order equations;342
13.3.1;11.3.1 Problem formulation;342
13.3.2;11.3.2 Scheme with weights;344
13.3.3;11.3.3 Additive schemes;346
13.3.4;11.3.4 More general problems;349
13.3.5;11.3.5 Problems of hydrodynamics;351
14;Bibliography;355
15;Index;369

Über Autor(innen)

Petr N. Vabishchevich, Russian Academy of Sciences, Moscow, Russia.