Additive Operator-Difference Schemes

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]