Additive Operator-Difference Schemes
Splitting Schemes
1st Edition
Published on 27. November 2013
XVI, 354 pages
978-3-11-032146-3 (ISBN)
Description
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.
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Petr N. Vabishchevich, Russian Academy of Sciences, Moscow, Russia.
Content
1 - Preface [Seite 5]
2 - Notation [Seite 15]
3 - 1 Introduction [Seite 17]
3.1 - 1.1 Numerical methods [Seite 17]
3.2 - 1.2 Additive operator-difference schemes [Seite 19]
3.3 - 1.3 The main results [Seite 22]
3.4 - 1.4 Contents of the book [Seite 26]
4 - 2 Stability of operator-difference schemes [Seite 30]
4.1 - 2.1 The Cauchy problem for an operator-differential equation [Seite 30]
4.1.1 - 2.1.1 Hilbert spaces [Seite 30]
4.1.2 - 2.1.2 Linear operators in a finite-dimensional space [Seite 32]
4.1.3 - 2.1.3 Operators in a finite-dimensional Hilbert space [Seite 33]
4.1.4 - 2.1.4 The Cauchy problem for an evolutionary equation of first order [Seite 35]
4.1.5 - 2.1.5 Systems of linear ordinary differential equations [Seite 36]
4.1.6 - 2.1.6 A boundary value problem for a one-dimensional parabolic equation [Seite 37]
4.1.7 - 2.1.7 Equations of second order [Seite 39]
4.2 - 2.2 Two-level schemes [Seite 40]
4.2.1 - 2.2.1 Key concepts [Seite 40]
4.2.2 - 2.2.2 Stability with respect to the initial data [Seite 42]
4.2.3 - 2.2.3 Stability with respect to the right-hand side [Seite 45]
4.2.4 - 2.2.4 Schemes with weights [Seite 47]
4.3 - 2.3 Three-level schemes [Seite 48]
4.3.1 - 2.3.1 Stability with respect to the initial data [Seite 48]
4.3.2 - 2.3.2 Reduction to a two-level scheme [Seite 50]
4.3.3 - 2.3.3 P-stability of three-level schemes [Seite 52]
4.3.4 - 2.3.4 Estimates in simpler norms [Seite 54]
4.3.5 - 2.3.5 Stability with respect to the right-hand side [Seite 56]
4.3.6 - 2.3.6 Schemes with weights for equations of first order [Seite 56]
4.3.7 - 2.3.7 Schemes with weights for equations of second order [Seite 58]
4.4 - 2.4 Stability in finite-dimensional Banach spaces [Seite 59]
4.4.1 - 2.4.1 The Cauchy problem for a system of ordinary differential equations [Seite 59]
4.4.2 - 2.4.2 Scheme with weights [Seite 61]
4.4.3 - 2.4.3 Difference schemes for a one-dimensional parabolic equation [Seite 63]
4.5 - 2.5 Stability of projection-difference schemes [Seite 63]
4.5.1 - 2.5.1 Preliminary observations [Seite 64]
4.5.2 - 2.5.2 Stability of finite element techniques [Seite 65]
4.5.3 - 2.5.3 Stability of projection-difference schemes [Seite 67]
4.5.4 - 2.5.4 Conditions for -stability of projection-difference schemes [Seite 69]
4.5.5 - 2.5.5 Schemes with weights [Seite 71]
4.5.6 - 2.5.6 Stability with respect to the right-hand side [Seite 73]
4.5.7 - 2.5.7 Stability of three-level schemes with respect to the initial data [Seite 75]
4.5.8 - 2.5.8 Stability with respect to the right-hand side [Seite 76]
4.5.9 - 2.5.9 Schemes for an equation of first order [Seite 77]
5 - 3 Operator splitting [Seite 79]
5.1 - 3.1 Time-dependent problems of convection-diffusion [Seite 79]
5.1.1 - 3.1.1 Differential problem [Seite 79]
5.1.2 - 3.1.2 Semi-discrete problem [Seite 84]
5.1.3 - 3.1.3 Two-level schemes [Seite 86]
5.2 - 3.2 Splitting operators in convection-diffusion problems [Seite 93]
5.2.1 - 3.2.1 Splitting with respect to spatial variables [Seite 93]
5.2.2 - 3.2.2 Splitting with respect to physical processes [Seite 94]
5.2.3 - 3.2.3 Schemes for problems with an operator semibounded from below [Seite 96]
5.3 - 3.3 Domain decomposition methods [Seite 98]
5.3.1 - 3.3.1 Preliminaries [Seite 98]
5.3.2 - 3.3.2 Model boundary value problems [Seite 101]
5.3.3 - 3.3.3 Standard finite difference approximations [Seite 103]
5.3.4 - 3.3.4 Domain decomposition [Seite 107]
5.3.5 - 3.3.5 Problems with non-self-adjoint operators [Seite 114]
5.4 - 3.4 Difference schemes for time-dependent vector problems [Seite 117]
5.4.1 - 3.4.1 Preliminary discussions [Seite 117]
5.4.2 - 3.4.2 Statement of the problem [Seite 118]
5.4.3 - 3.4.3 Estimates for the solution of differential problems [Seite 120]
5.4.4 - 3.4.4 Approximation in space [Seite 122]
5.4.5 - 3.4.5 Schemes with weights [Seite 124]
5.4.6 - 3.4.6 Alternating triangle method [Seite 125]
5.5 - 3.5 Problems of hydrodynamics of an incompressible fluid [Seite 128]
5.5.1 - 3.5.1 Differential problem [Seite 128]
5.5.2 - 3.5.2 Discretization in space [Seite 130]
5.5.3 - 3.5.3 Peculiarities of hydrodynamic equations written in the primitive variables [Seite 133]
5.5.4 - 3.5.4 A priori estimate for the differential problem [Seite 134]
5.5.5 - 3.5.5 Approximation in space [Seite 135]
5.5.6 - 3.5.6 Additive difference schemes [Seite 137]
6 - 4 Additive schemes of two-component splitting [Seite 139]
6.1 - 4.1 Alternating direction implicit schemes [Seite 139]
6.1.1 - 4.1.1 Problem formulation [Seite 139]
6.1.2 - 4.1.2 The Peaceman-Rachford scheme [Seite 140]
6.1.3 - 4.1.3 Stability of the Peaceman-Rachford scheme [Seite 141]
6.1.4 - 4.1.4 Accuracy of the Peaceman-Rachford scheme [Seite 142]
6.1.5 - 4.1.5 Another ADI scheme [Seite 143]
6.2 - 4.2 Factorized schemes [Seite 143]
6.2.1 - 4.2.1 General considerations [Seite 144]
6.2.2 - 4.2.2 ADI methods as factorized schemes [Seite 144]
6.2.3 - 4.2.3 Stability and accuracy of factorized schemes [Seite 145]
6.2.4 - 4.2.4 Regularization principle for constructing factorized schemes [Seite 147]
6.2.5 - 4.2.5 Factorized schemes of multicomponent splitting [Seite 149]
6.3 - 4.3 Alternating triangle method [Seite 150]
6.3.1 - 4.3.1 General description of the alternating triangle method [Seite 151]
6.3.2 - 4.3.2 Investigation of stability and convergence [Seite 152]
6.3.3 - 4.3.3 Three-level additive schemes [Seite 153]
6.3.4 - 4.3.4 Problems with non-self-adjoint operators [Seite 155]
6.4 - 4.4 Equations of second order [Seite 156]
6.4.1 - 4.4.1 Model problem [Seite 157]
6.4.2 - 4.4.2 Factorized schemes [Seite 158]
6.4.3 - 4.4.3 Schemes of the alternating triangle method [Seite 159]
7 - 5 Schemes of summarized approximation [Seite 160]
7.1 - 5.1 Additive formulations of differential problems [Seite 160]
7.1.1 - 5.1.1 Model problem [Seite 160]
7.1.2 - 5.1.2 Intermediate problems [Seite 161]
7.1.3 - 5.1.3 Summarized approximation concept [Seite 163]
7.1.4 - 5.1.4 Schemes of the second-order summarized approximation [Seite 164]
7.2 - 5.2 Investigation of schemes of summarized approximation [Seite 166]
7.2.1 - 5.2.1 Schemes of componentwise splitting [Seite 166]
7.2.2 - 5.2.2 Estimates for the intermediate problem solutions [Seite 167]
7.2.3 - 5.2.3 Stability of componentwise splitting schemes [Seite 169]
7.2.4 - 5.2.4 Convergence of componentwise splitting schemes [Seite 170]
7.2.5 - 5.2.5 Convergence of additive schemes in Banach spaces [Seite 171]
7.3 - 5.3 Additively averaged schemes [Seite 172]
7.3.1 - 5.3.1 Differential problem [Seite 172]
7.3.2 - 5.3.2 Additive schemes [Seite 173]
7.3.3 - 5.3.3 Stability of additively averaged schemes [Seite 174]
7.4 - 5.4 Other variants of componentwise splitting schemes [Seite 176]
7.4.1 - 5.4.1 Fully implicit additive schemes [Seite 176]
7.4.2 - 5.4.2 ADI methods as additive schemes [Seite 177]
7.4.3 - 5.4.3 Additive schemes with second-order accuracy [Seite 178]
7.4.4 - 5.4.4 Convergence of higher-order schemes [Seite 179]
8 - 6 Regularized additive schemes [Seite 183]
8.1 - 6.1 Multiplicative regularization of difference schemes [Seite 183]
8.1.1 - 6.1.1 Regularization principle for difference schemes [Seite 183]
8.1.2 - 6.1.2 Additive regularization [Seite 184]
8.1.3 - 6.1.3 Multiplicative regularization [Seite 186]
8.2 - 6.2 Multiplicative regularization of additive schemes [Seite 187]
8.2.1 - 6.2.1 The Cauchy problem for a first-order equation [Seite 187]
8.2.2 - 6.2.2 Regularization of additive schemes [Seite 188]
8.2.3 - 6.2.3 Stability and convergence [Seite 189]
8.2.4 - 6.2.4 Regularized and additively averaged schemes [Seite 191]
8.3 - 6.3 Schemes of higher-order accuracy [Seite 192]
8.3.1 - 6.3.1 Statement of the problem [Seite 192]
8.3.2 - 6.3.2 Explicit three-level scheme [Seite 193]
8.3.3 - 6.3.3 Regularized schemes [Seite 194]
8.3.4 - 6.3.4 Additively averaged scheme [Seite 195]
8.4 - 6.4 Regularized schemes for equations of second order [Seite 196]
8.4.1 - 6.4.1 Model problem [Seite 196]
8.4.2 - 6.4.2 Regularized scheme [Seite 197]
8.4.3 - 6.4.3 Additively averaged schemes for equations of second order [Seite 198]
8.5 - 6.5 Regularized schemes with general regularizers [Seite 199]
8.5.1 - 6.5.1 General regularizers [Seite 199]
8.5.2 - 6.5.2 Additive schemes with a general-form regularizer [Seite 201]
8.5.3 - 6.5.3 Factorized additive schemes [Seite 202]
8.5.4 - 6.5.4 Generalizations [Seite 203]
9 - 7 Schemes based on approximations of a transition operator [Seite 206]
9.1 - 7.1 Operator-difference schemes [Seite 206]
9.1.1 - 7.1.1 Operator-differential problem [Seite 206]
9.1.2 - 7.1.2 Difference approximations in time [Seite 207]
9.1.3 - 7.1.3 SM-stable schemes for problems with a self-adjoint operator [Seite 210]
9.1.4 - 7.1.4 Factorized SM-stable two-level schemes [Seite 215]
9.1.5 - 7.1.5 Problems with a skew-symmetric operator [Seite 219]
9.2 - 7.2 Additive schemes with a multiplicative transition operator [Seite 220]
9.2.1 - 7.2.1 Operator-differential problems [Seite 220]
9.2.2 - 7.2.2 Componentwise splitting schemes [Seite 222]
9.3 - 7.3 Splitting schemes with an additive transition operator [Seite 224]
9.3.1 - 7.3.1 Additive approximation of a transition operator [Seite 225]
9.3.2 - 7.3.2 Additive schemes [Seite 225]
9.3.3 - 7.3.3 Regularized additive schemes [Seite 227]
9.4 - 7.4 Further additive schemes [Seite 227]
9.4.1 - 7.4.1 Schemes of the second order [Seite 228]
9.4.2 - 7.4.2 Factorized schemes [Seite 229]
9.4.3 - 7.4.3 Inhomogeneous approximation of a transition operator [Seite 230]
9.4.4 - 7.4.4 Schemes of higher-order approximation [Seite 231]
10 - 8 Vector additive schemes [Seite 234]
10.1 - 8.1 Vector schemes for first-order equations [Seite 234]
10.1.1 - 8.1.1 Vector differential problem [Seite 234]
10.1.2 - 8.1.2 Stability of vector additive schemes [Seite 236]
10.1.3 - 8.1.3 Stability with respect to the right-hand side [Seite 239]
10.2 - 8.2 Stability of vector additive schemes in Banach spaces [Seite 240]
10.2.1 - 8.2.1 Problem formulation [Seite 240]
10.2.2 - 8.2.2 Vector additive scheme [Seite 241]
10.2.3 - 8.2.3 Study on stability [Seite 242]
10.3 - 8.3 Schemes of second-order accuracy [Seite 244]
10.3.1 - 8.3.1 Statement of the problem [Seite 244]
10.3.2 - 8.3.2 Three-level vector schemes [Seite 245]
10.3.3 - 8.3.3 Schemes of the alternating triangle method [Seite 247]
10.4 - 8.4 Vector schemes for equations of second order [Seite 248]
10.4.1 - 8.4.1 The Cauchy problem for a second-order equation [Seite 248]
10.4.2 - 8.4.2 Vector problem [Seite 250]
10.4.3 - 8.4.3 Scheme with weights [Seite 251]
10.4.4 - 8.4.4 Additive schemes [Seite 252]
10.4.5 - 8.4.5 Stability of additive schemes [Seite 254]
11 - 9 Iterative methods [Seite 256]
11.1 - 9.1 Basics of iterative methods [Seite 256]
11.1.1 - 9.1.1 Problem formulation [Seite 256]
11.1.2 - 9.1.2 Simple iteration method [Seite 258]
11.1.3 - 9.1.3 The Chebyshev iterative method [Seite 259]
11.1.4 - 9.1.4 Two-level variation-type methods [Seite 260]
11.1.5 - 9.1.5 Conjugate gradient method [Seite 261]
11.2 - 9.2 Iterative alternating direction method [Seite 262]
11.2.1 - 9.2.1 Iterative method with two-component splitting [Seite 262]
11.2.2 - 9.2.2 Convergence study [Seite 263]
11.2.3 - 9.2.3 Modified iterative method of alternating directions [Seite 265]
11.2.4 - 9.2.4 Multicomponent splitting [Seite 266]
11.3 - 9.3 Iterative alternating triangle method [Seite 268]
11.3.1 - 9.3.1 Iterative method [Seite 268]
11.3.2 - 9.3.2 Convergence rate [Seite 269]
11.3.3 - 9.3.3 Modified iterative method of alternating triangles [Seite 271]
11.4 - 9.4 Iterative cluster aggregation methods [Seite 271]
11.4.1 - 9.4.1 Transition to a system of equations [Seite 272]
11.4.2 - 9.4.2 Iterative method [Seite 273]
11.4.3 - 9.4.3 Parallel variant [Seite 275]
11.4.4 - 9.4.4 Aggregation of unknowns [Seite 276]
12 - 10 Splitting of the operator at the time derivative [Seite 279]
12.1 - 10.1 Schemes with splitting of the operator at the time derivative [Seite 279]
12.1.1 - 10.1.1 Preliminary discussions [Seite 279]
12.1.2 - 10.1.2 Statement of the problem [Seite 280]
12.1.3 - 10.1.3 Vector problem [Seite 282]
12.1.4 - 10.1.4 Vector additive schemes [Seite 284]
12.1.5 - 10.1.5 Generalizations [Seite 288]
12.2 - 10.2 General splitting [Seite 288]
12.2.1 - 10.2.1 Preliminary discussions [Seite 289]
12.2.2 - 10.2.2 Problem formulation [Seite 290]
12.2.3 - 10.2.3 Scheme with weights [Seite 292]
12.2.4 - 10.2.4 Schemes with a diagonal operator [Seite 294]
12.2.5 - 10.2.5 The more general problem [Seite 295]
12.3 - 10.3 Explicit-implicit splitting schemes [Seite 298]
12.3.1 - 10.3.1 Introduction [Seite 298]
12.3.2 - 10.3.2 Boundary value problems for systems of equations [Seite 299]
12.3.3 - 10.3.3 Schemes with a diagonal operator [Seite 301]
12.3.4 - 10.3.4 General case [Seite 305]
13 - 11 Equations with pairwise adjoint operators [Seite 307]
13.1 - 11.1 Splitting schemes for a system of equations [Seite 307]
13.1.1 - 11.1.1 Preliminary discussions [Seite 308]
13.1.2 - 11.1.2 Statement of the problem [Seite 309]
13.1.3 - 11.1.3 A priori estimates [Seite 311]
13.1.4 - 11.1.4 Schemes with weights [Seite 315]
13.1.5 - 11.1.5 Splitting schemes to find the p-th component of the solution [Seite 319]
13.1.6 - 11.1.6 Additive schemes for systems of equations [Seite 322]
13.2 - 11.2 Additive schemes for a system of first-order equations [Seite 326]
13.2.1 - 11.2.1 Statement of the problem [Seite 326]
13.2.2 - 11.2.2 Examples [Seite 329]
13.2.3 - 11.2.3 Schemes with weights [Seite 332]
13.2.4 - 11.2.4 Explicit-implicit schemes [Seite 334]
13.2.5 - 11.2.5 Additive schemes of componentwise splitting [Seite 338]
13.2.6 - 11.2.6 Regularized additive schemes [Seite 340]
13.3 - 11.3 Another class of systems of first-order equations [Seite 342]
13.3.1 - 11.3.1 Problem formulation [Seite 342]
13.3.2 - 11.3.2 Scheme with weights [Seite 344]
13.3.3 - 11.3.3 Additive schemes [Seite 346]
13.3.4 - 11.3.4 More general problems [Seite 349]
13.3.5 - 11.3.5 Problems of hydrodynamics [Seite 351]
14 - Bibliography [Seite 355]
15 - Index [Seite 369]
2 - Notation [Seite 15]
3 - 1 Introduction [Seite 17]
3.1 - 1.1 Numerical methods [Seite 17]
3.2 - 1.2 Additive operator-difference schemes [Seite 19]
3.3 - 1.3 The main results [Seite 22]
3.4 - 1.4 Contents of the book [Seite 26]
4 - 2 Stability of operator-difference schemes [Seite 30]
4.1 - 2.1 The Cauchy problem for an operator-differential equation [Seite 30]
4.1.1 - 2.1.1 Hilbert spaces [Seite 30]
4.1.2 - 2.1.2 Linear operators in a finite-dimensional space [Seite 32]
4.1.3 - 2.1.3 Operators in a finite-dimensional Hilbert space [Seite 33]
4.1.4 - 2.1.4 The Cauchy problem for an evolutionary equation of first order [Seite 35]
4.1.5 - 2.1.5 Systems of linear ordinary differential equations [Seite 36]
4.1.6 - 2.1.6 A boundary value problem for a one-dimensional parabolic equation [Seite 37]
4.1.7 - 2.1.7 Equations of second order [Seite 39]
4.2 - 2.2 Two-level schemes [Seite 40]
4.2.1 - 2.2.1 Key concepts [Seite 40]
4.2.2 - 2.2.2 Stability with respect to the initial data [Seite 42]
4.2.3 - 2.2.3 Stability with respect to the right-hand side [Seite 45]
4.2.4 - 2.2.4 Schemes with weights [Seite 47]
4.3 - 2.3 Three-level schemes [Seite 48]
4.3.1 - 2.3.1 Stability with respect to the initial data [Seite 48]
4.3.2 - 2.3.2 Reduction to a two-level scheme [Seite 50]
4.3.3 - 2.3.3 P-stability of three-level schemes [Seite 52]
4.3.4 - 2.3.4 Estimates in simpler norms [Seite 54]
4.3.5 - 2.3.5 Stability with respect to the right-hand side [Seite 56]
4.3.6 - 2.3.6 Schemes with weights for equations of first order [Seite 56]
4.3.7 - 2.3.7 Schemes with weights for equations of second order [Seite 58]
4.4 - 2.4 Stability in finite-dimensional Banach spaces [Seite 59]
4.4.1 - 2.4.1 The Cauchy problem for a system of ordinary differential equations [Seite 59]
4.4.2 - 2.4.2 Scheme with weights [Seite 61]
4.4.3 - 2.4.3 Difference schemes for a one-dimensional parabolic equation [Seite 63]
4.5 - 2.5 Stability of projection-difference schemes [Seite 63]
4.5.1 - 2.5.1 Preliminary observations [Seite 64]
4.5.2 - 2.5.2 Stability of finite element techniques [Seite 65]
4.5.3 - 2.5.3 Stability of projection-difference schemes [Seite 67]
4.5.4 - 2.5.4 Conditions for -stability of projection-difference schemes [Seite 69]
4.5.5 - 2.5.5 Schemes with weights [Seite 71]
4.5.6 - 2.5.6 Stability with respect to the right-hand side [Seite 73]
4.5.7 - 2.5.7 Stability of three-level schemes with respect to the initial data [Seite 75]
4.5.8 - 2.5.8 Stability with respect to the right-hand side [Seite 76]
4.5.9 - 2.5.9 Schemes for an equation of first order [Seite 77]
5 - 3 Operator splitting [Seite 79]
5.1 - 3.1 Time-dependent problems of convection-diffusion [Seite 79]
5.1.1 - 3.1.1 Differential problem [Seite 79]
5.1.2 - 3.1.2 Semi-discrete problem [Seite 84]
5.1.3 - 3.1.3 Two-level schemes [Seite 86]
5.2 - 3.2 Splitting operators in convection-diffusion problems [Seite 93]
5.2.1 - 3.2.1 Splitting with respect to spatial variables [Seite 93]
5.2.2 - 3.2.2 Splitting with respect to physical processes [Seite 94]
5.2.3 - 3.2.3 Schemes for problems with an operator semibounded from below [Seite 96]
5.3 - 3.3 Domain decomposition methods [Seite 98]
5.3.1 - 3.3.1 Preliminaries [Seite 98]
5.3.2 - 3.3.2 Model boundary value problems [Seite 101]
5.3.3 - 3.3.3 Standard finite difference approximations [Seite 103]
5.3.4 - 3.3.4 Domain decomposition [Seite 107]
5.3.5 - 3.3.5 Problems with non-self-adjoint operators [Seite 114]
5.4 - 3.4 Difference schemes for time-dependent vector problems [Seite 117]
5.4.1 - 3.4.1 Preliminary discussions [Seite 117]
5.4.2 - 3.4.2 Statement of the problem [Seite 118]
5.4.3 - 3.4.3 Estimates for the solution of differential problems [Seite 120]
5.4.4 - 3.4.4 Approximation in space [Seite 122]
5.4.5 - 3.4.5 Schemes with weights [Seite 124]
5.4.6 - 3.4.6 Alternating triangle method [Seite 125]
5.5 - 3.5 Problems of hydrodynamics of an incompressible fluid [Seite 128]
5.5.1 - 3.5.1 Differential problem [Seite 128]
5.5.2 - 3.5.2 Discretization in space [Seite 130]
5.5.3 - 3.5.3 Peculiarities of hydrodynamic equations written in the primitive variables [Seite 133]
5.5.4 - 3.5.4 A priori estimate for the differential problem [Seite 134]
5.5.5 - 3.5.5 Approximation in space [Seite 135]
5.5.6 - 3.5.6 Additive difference schemes [Seite 137]
6 - 4 Additive schemes of two-component splitting [Seite 139]
6.1 - 4.1 Alternating direction implicit schemes [Seite 139]
6.1.1 - 4.1.1 Problem formulation [Seite 139]
6.1.2 - 4.1.2 The Peaceman-Rachford scheme [Seite 140]
6.1.3 - 4.1.3 Stability of the Peaceman-Rachford scheme [Seite 141]
6.1.4 - 4.1.4 Accuracy of the Peaceman-Rachford scheme [Seite 142]
6.1.5 - 4.1.5 Another ADI scheme [Seite 143]
6.2 - 4.2 Factorized schemes [Seite 143]
6.2.1 - 4.2.1 General considerations [Seite 144]
6.2.2 - 4.2.2 ADI methods as factorized schemes [Seite 144]
6.2.3 - 4.2.3 Stability and accuracy of factorized schemes [Seite 145]
6.2.4 - 4.2.4 Regularization principle for constructing factorized schemes [Seite 147]
6.2.5 - 4.2.5 Factorized schemes of multicomponent splitting [Seite 149]
6.3 - 4.3 Alternating triangle method [Seite 150]
6.3.1 - 4.3.1 General description of the alternating triangle method [Seite 151]
6.3.2 - 4.3.2 Investigation of stability and convergence [Seite 152]
6.3.3 - 4.3.3 Three-level additive schemes [Seite 153]
6.3.4 - 4.3.4 Problems with non-self-adjoint operators [Seite 155]
6.4 - 4.4 Equations of second order [Seite 156]
6.4.1 - 4.4.1 Model problem [Seite 157]
6.4.2 - 4.4.2 Factorized schemes [Seite 158]
6.4.3 - 4.4.3 Schemes of the alternating triangle method [Seite 159]
7 - 5 Schemes of summarized approximation [Seite 160]
7.1 - 5.1 Additive formulations of differential problems [Seite 160]
7.1.1 - 5.1.1 Model problem [Seite 160]
7.1.2 - 5.1.2 Intermediate problems [Seite 161]
7.1.3 - 5.1.3 Summarized approximation concept [Seite 163]
7.1.4 - 5.1.4 Schemes of the second-order summarized approximation [Seite 164]
7.2 - 5.2 Investigation of schemes of summarized approximation [Seite 166]
7.2.1 - 5.2.1 Schemes of componentwise splitting [Seite 166]
7.2.2 - 5.2.2 Estimates for the intermediate problem solutions [Seite 167]
7.2.3 - 5.2.3 Stability of componentwise splitting schemes [Seite 169]
7.2.4 - 5.2.4 Convergence of componentwise splitting schemes [Seite 170]
7.2.5 - 5.2.5 Convergence of additive schemes in Banach spaces [Seite 171]
7.3 - 5.3 Additively averaged schemes [Seite 172]
7.3.1 - 5.3.1 Differential problem [Seite 172]
7.3.2 - 5.3.2 Additive schemes [Seite 173]
7.3.3 - 5.3.3 Stability of additively averaged schemes [Seite 174]
7.4 - 5.4 Other variants of componentwise splitting schemes [Seite 176]
7.4.1 - 5.4.1 Fully implicit additive schemes [Seite 176]
7.4.2 - 5.4.2 ADI methods as additive schemes [Seite 177]
7.4.3 - 5.4.3 Additive schemes with second-order accuracy [Seite 178]
7.4.4 - 5.4.4 Convergence of higher-order schemes [Seite 179]
8 - 6 Regularized additive schemes [Seite 183]
8.1 - 6.1 Multiplicative regularization of difference schemes [Seite 183]
8.1.1 - 6.1.1 Regularization principle for difference schemes [Seite 183]
8.1.2 - 6.1.2 Additive regularization [Seite 184]
8.1.3 - 6.1.3 Multiplicative regularization [Seite 186]
8.2 - 6.2 Multiplicative regularization of additive schemes [Seite 187]
8.2.1 - 6.2.1 The Cauchy problem for a first-order equation [Seite 187]
8.2.2 - 6.2.2 Regularization of additive schemes [Seite 188]
8.2.3 - 6.2.3 Stability and convergence [Seite 189]
8.2.4 - 6.2.4 Regularized and additively averaged schemes [Seite 191]
8.3 - 6.3 Schemes of higher-order accuracy [Seite 192]
8.3.1 - 6.3.1 Statement of the problem [Seite 192]
8.3.2 - 6.3.2 Explicit three-level scheme [Seite 193]
8.3.3 - 6.3.3 Regularized schemes [Seite 194]
8.3.4 - 6.3.4 Additively averaged scheme [Seite 195]
8.4 - 6.4 Regularized schemes for equations of second order [Seite 196]
8.4.1 - 6.4.1 Model problem [Seite 196]
8.4.2 - 6.4.2 Regularized scheme [Seite 197]
8.4.3 - 6.4.3 Additively averaged schemes for equations of second order [Seite 198]
8.5 - 6.5 Regularized schemes with general regularizers [Seite 199]
8.5.1 - 6.5.1 General regularizers [Seite 199]
8.5.2 - 6.5.2 Additive schemes with a general-form regularizer [Seite 201]
8.5.3 - 6.5.3 Factorized additive schemes [Seite 202]
8.5.4 - 6.5.4 Generalizations [Seite 203]
9 - 7 Schemes based on approximations of a transition operator [Seite 206]
9.1 - 7.1 Operator-difference schemes [Seite 206]
9.1.1 - 7.1.1 Operator-differential problem [Seite 206]
9.1.2 - 7.1.2 Difference approximations in time [Seite 207]
9.1.3 - 7.1.3 SM-stable schemes for problems with a self-adjoint operator [Seite 210]
9.1.4 - 7.1.4 Factorized SM-stable two-level schemes [Seite 215]
9.1.5 - 7.1.5 Problems with a skew-symmetric operator [Seite 219]
9.2 - 7.2 Additive schemes with a multiplicative transition operator [Seite 220]
9.2.1 - 7.2.1 Operator-differential problems [Seite 220]
9.2.2 - 7.2.2 Componentwise splitting schemes [Seite 222]
9.3 - 7.3 Splitting schemes with an additive transition operator [Seite 224]
9.3.1 - 7.3.1 Additive approximation of a transition operator [Seite 225]
9.3.2 - 7.3.2 Additive schemes [Seite 225]
9.3.3 - 7.3.3 Regularized additive schemes [Seite 227]
9.4 - 7.4 Further additive schemes [Seite 227]
9.4.1 - 7.4.1 Schemes of the second order [Seite 228]
9.4.2 - 7.4.2 Factorized schemes [Seite 229]
9.4.3 - 7.4.3 Inhomogeneous approximation of a transition operator [Seite 230]
9.4.4 - 7.4.4 Schemes of higher-order approximation [Seite 231]
10 - 8 Vector additive schemes [Seite 234]
10.1 - 8.1 Vector schemes for first-order equations [Seite 234]
10.1.1 - 8.1.1 Vector differential problem [Seite 234]
10.1.2 - 8.1.2 Stability of vector additive schemes [Seite 236]
10.1.3 - 8.1.3 Stability with respect to the right-hand side [Seite 239]
10.2 - 8.2 Stability of vector additive schemes in Banach spaces [Seite 240]
10.2.1 - 8.2.1 Problem formulation [Seite 240]
10.2.2 - 8.2.2 Vector additive scheme [Seite 241]
10.2.3 - 8.2.3 Study on stability [Seite 242]
10.3 - 8.3 Schemes of second-order accuracy [Seite 244]
10.3.1 - 8.3.1 Statement of the problem [Seite 244]
10.3.2 - 8.3.2 Three-level vector schemes [Seite 245]
10.3.3 - 8.3.3 Schemes of the alternating triangle method [Seite 247]
10.4 - 8.4 Vector schemes for equations of second order [Seite 248]
10.4.1 - 8.4.1 The Cauchy problem for a second-order equation [Seite 248]
10.4.2 - 8.4.2 Vector problem [Seite 250]
10.4.3 - 8.4.3 Scheme with weights [Seite 251]
10.4.4 - 8.4.4 Additive schemes [Seite 252]
10.4.5 - 8.4.5 Stability of additive schemes [Seite 254]
11 - 9 Iterative methods [Seite 256]
11.1 - 9.1 Basics of iterative methods [Seite 256]
11.1.1 - 9.1.1 Problem formulation [Seite 256]
11.1.2 - 9.1.2 Simple iteration method [Seite 258]
11.1.3 - 9.1.3 The Chebyshev iterative method [Seite 259]
11.1.4 - 9.1.4 Two-level variation-type methods [Seite 260]
11.1.5 - 9.1.5 Conjugate gradient method [Seite 261]
11.2 - 9.2 Iterative alternating direction method [Seite 262]
11.2.1 - 9.2.1 Iterative method with two-component splitting [Seite 262]
11.2.2 - 9.2.2 Convergence study [Seite 263]
11.2.3 - 9.2.3 Modified iterative method of alternating directions [Seite 265]
11.2.4 - 9.2.4 Multicomponent splitting [Seite 266]
11.3 - 9.3 Iterative alternating triangle method [Seite 268]
11.3.1 - 9.3.1 Iterative method [Seite 268]
11.3.2 - 9.3.2 Convergence rate [Seite 269]
11.3.3 - 9.3.3 Modified iterative method of alternating triangles [Seite 271]
11.4 - 9.4 Iterative cluster aggregation methods [Seite 271]
11.4.1 - 9.4.1 Transition to a system of equations [Seite 272]
11.4.2 - 9.4.2 Iterative method [Seite 273]
11.4.3 - 9.4.3 Parallel variant [Seite 275]
11.4.4 - 9.4.4 Aggregation of unknowns [Seite 276]
12 - 10 Splitting of the operator at the time derivative [Seite 279]
12.1 - 10.1 Schemes with splitting of the operator at the time derivative [Seite 279]
12.1.1 - 10.1.1 Preliminary discussions [Seite 279]
12.1.2 - 10.1.2 Statement of the problem [Seite 280]
12.1.3 - 10.1.3 Vector problem [Seite 282]
12.1.4 - 10.1.4 Vector additive schemes [Seite 284]
12.1.5 - 10.1.5 Generalizations [Seite 288]
12.2 - 10.2 General splitting [Seite 288]
12.2.1 - 10.2.1 Preliminary discussions [Seite 289]
12.2.2 - 10.2.2 Problem formulation [Seite 290]
12.2.3 - 10.2.3 Scheme with weights [Seite 292]
12.2.4 - 10.2.4 Schemes with a diagonal operator [Seite 294]
12.2.5 - 10.2.5 The more general problem [Seite 295]
12.3 - 10.3 Explicit-implicit splitting schemes [Seite 298]
12.3.1 - 10.3.1 Introduction [Seite 298]
12.3.2 - 10.3.2 Boundary value problems for systems of equations [Seite 299]
12.3.3 - 10.3.3 Schemes with a diagonal operator [Seite 301]
12.3.4 - 10.3.4 General case [Seite 305]
13 - 11 Equations with pairwise adjoint operators [Seite 307]
13.1 - 11.1 Splitting schemes for a system of equations [Seite 307]
13.1.1 - 11.1.1 Preliminary discussions [Seite 308]
13.1.2 - 11.1.2 Statement of the problem [Seite 309]
13.1.3 - 11.1.3 A priori estimates [Seite 311]
13.1.4 - 11.1.4 Schemes with weights [Seite 315]
13.1.5 - 11.1.5 Splitting schemes to find the p-th component of the solution [Seite 319]
13.1.6 - 11.1.6 Additive schemes for systems of equations [Seite 322]
13.2 - 11.2 Additive schemes for a system of first-order equations [Seite 326]
13.2.1 - 11.2.1 Statement of the problem [Seite 326]
13.2.2 - 11.2.2 Examples [Seite 329]
13.2.3 - 11.2.3 Schemes with weights [Seite 332]
13.2.4 - 11.2.4 Explicit-implicit schemes [Seite 334]
13.2.5 - 11.2.5 Additive schemes of componentwise splitting [Seite 338]
13.2.6 - 11.2.6 Regularized additive schemes [Seite 340]
13.3 - 11.3 Another class of systems of first-order equations [Seite 342]
13.3.1 - 11.3.1 Problem formulation [Seite 342]
13.3.2 - 11.3.2 Scheme with weights [Seite 344]
13.3.3 - 11.3.3 Additive schemes [Seite 346]
13.3.4 - 11.3.4 More general problems [Seite 349]
13.3.5 - 11.3.5 Problems of hydrodynamics [Seite 351]
14 - Bibliography [Seite 355]
15 - Index [Seite 369]
