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

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.