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John Strikwerda — Finite difference schemes and partial differential equations
John Strikwerda — Finite difference schemes and partial differential equations

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Название: Finite difference schemes and partial differential equations

Автор: John Strikwerda

Аннотация:

This book provides a unified and accessible introduction to the basic theory of finite difference schemes applied to the numerical solution of partial differential equations. Originally published in 1989, its objective remains to clearly present the basic methods necessary to perform finite difference schemes and to understand the theory underlying these schemes. This is one of the few texts in the field to not only present the theory of stability in a rigorous and clear manner but also to discuss the theory of initial-boundary value problems in relation to finite difference schemes. In this updated edition the notion of a stability domain is now included in the definition of stability and is more prevalent throughout the book. The author has also added many new figures and tables to clarify important concepts and illustrate the properties of finite difference schemes.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2

Год издания: 2007

Количество страниц: 450

Добавлена в каталог: 18.10.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$L^{2}$ norm on grid      29 39
$L^{2}$ norm on real line      38
ADI methods      172—185 342
ADI methods for second-order equations      202
ADI methods with mixed derivatives      181
ADI methods, boundary conditions for      176
ADI methods, implementation of      177—180
ADI methods, stability      177
Admissible solution      291 301
Amplification factor      48
Amplification factor for multistep scheme      97 267
Amplification factor of second-order equations      193 271
Amplification matrix      166
Amplification polynomial      103 123 245 289
Analytic function      419
Artificial viscosity      161
Backward-time backward-space scheme      35
Backward-time central-space scheme for hyperbolic equations      35 57
Backward-time central-space scheme for the heat equation      147
Biharmonic equation      313
Block tridiagonal systems      90
boundary conditions      176
Boundary conditions for ADI schemes      176
Boundary conditions for elliptic equations      322
Boundary conditions for finite difference schemes      85 281
Boundary conditions for parabolic equations      152
Boundary conditions, analysis of      279
Box scheme      57 77
Box scheme, modified      78
Brownian motion      141
Cauchy — Riemann equations      313
Cauchy — Schwarz inequality      377
Cell Peclet number      159
Cell Reynolds number      159
CFL condition      34
Chapman — Kolmogorov equation      141
Characteristics      2
Characteristics for systems      9
Characteristics for variable coefficients      5
Characteristics, incoming      11
Characteristics, outgoing      11
Checkerboard ordering      359
Cholesky factorization      395
Cholesky factorization, preconditioning      395
Conjugate gradient method      377
Conjugate search directions      380
Conjugate transpose      401
Conservative polynomial      109
Consistent scheme      25
Consistently ordered matrix      357
Continuity of the solution on the data      205
Convection-diffusion equation      140 157
Convergence estimate      387
Convergence estimate, implementation      384
Convergence estimates for nonsmooth initial functions      252—258
Convergence estimates for parabolic equations      259—261
Convergence estimates for smooth functions      235—246
Convergent scheme      23 262
Coordinate changes and schemes      335
Courant — Friedrichs — Lewy condition      34
Crank-Nicolson scheme for heat equation      147
Crank-Nicolson scheme for heat equation, nondissippative      151
Crank-Nicolson scheme for hyperbolic, adding dissipation to      123
Crank-Nicolson scheme for hyperbolic, boundary conditions      86 292
Crank-Nicolson scheme for hyperbolic, equations      63
Crank-Nicolson scheme for hyperbolic, modified      85
Crank-Nicolson scheme for hyperbolic, order of accuracy      68
Crank-Nicolson scheme for hyperbolic, solution of      88
Crank-Nicolson scheme for hyperbolic, stability of      77
Diagonalizable matrix      3
Diagonally dominant matrices      349
Difference calculus      78—82
Dirichlet boundary condition      145 152 176 199 311
dispersion      125
Dispersion in higher dimensions      202
Dispersive equation      190
Dissipation      122
Dissipation and smoothness of the solution      149
Dissipation for parabolic schemes      146
Dissipation, adding to nondissipative schemes      123
Dissipation, convergence estimates      259
Douglas — Rachford method      175
Du Fort — Frankel scheme      148 268
Duhamel’s principle      32 225 262
Dynamic stability      59
D’Yakonov scheme      175
Efficiency of higher order schemes      101 181
Eigenvalue of a matrix      403
Eigenvalue of a matrix, semisimple      403
Eigenvector      403
Eigenvector, generalized      403
Elliptic equations, boundary conditions      322
Elliptic equations, differentiability of the solution      314
Elliptic equations, discontinuous boundary data      322
Elliptic equations, regularity estimates      315
Energy method      145 191
Envelope of a wave packet      130
Euler backward scheme      57 75
Euler — Bernoulli equation      190
Euler — Bernoulli equation and piecewise smooth functions      256—258
Euler — Bernoulli equation, evaluation operator      242
Euler — Bernoulli equation, scheme for      195 200
Explicit schemes, definition      34
Exponential of a matrix      214 406
Faber — Krahn inequality      358
Factor space      367
Finite difference grid      16
Finite Fourier transform      46
Five-point (discrete) Laplacian      325
Fokker — Planck equations      140
Forward-time backward-space scheme      17 47
Forward-time central-space scheme      17
Forward-time central-space scheme and smoothing      55
Forward-time central-space scheme for hyperbolic equations      17 51
Forward-time central-space scheme for the heat equation      145
Forward-time forward-space scheme      17 27
Fourier analysis on the integers Z      38
Fourier analysis on the real line      37
Fourier analysis, differentiability of functions      42
Fourier inversion formula on the grid      38
Fourier inversion formula on the integers      38
Fourier inversion formula on the real line      37
Fourier inversion formula, multidimensional      44
Fourier series      38
Fourier transform in higher dimensions      44
Fourier transform of derivatives      42
Fourier transform on the integers      38
Fourier transform on the real line      37
Fourth-order accurate approximations of first derivative      79 80
Fourth-order accurate approximations of second derivative      80
Fourth-order accurate nine-point Laplacian      328
Frozen coefficient problems      59 276
Function spaces, $L^{1}(R), L^{2}(R), L^{\infty}(R)$      417
Gauss — Seidel algorithm      340
Gauss — Seidel algorithm and diagonally dominate matrices      349—351
Gauss — Seidel algorithm, analysis of      347
Gauss — Seidel algorithm, iteration matrix for      345
Gaussian elimination      88 89 339
Gaussian elimination, grid      16
Group velocity      130 190 248
Gustaffson-Kreiss-Sundstrom-Osher (GKSO) method      288 309
Harmonic functions      311 319 419
Heat equation      137
Hermitian matrix      226 230 408
Hurwitz polynomial      117
Hyperbolic equation      1
Hyperbolic equation with variable coefficients      5
Hyperbolic equation, differentiability of solutions      2
Hyperbolic systems      3 166 217
Hyperbolic systems, weakly      221
Implementation of iterative methods      346
Implicit schemes      34
Implicit schemes, solution of      88
Initial value problem for heat equation      137
Initial value problem for one-way wave equation      1
Initial value problem for second-ordcr equations      187
Initial value problem, analysis of      205—225
Initial-boundary value problems      275—310
Initial-boundary value problems for hyperbolic schemes      291
Initial-boundary value problems for parabolic schemes      292
Initial-boundary value problems for partial differential equations      300
Initialization for leapfrog scheme      18
Initialization of multistep schemes      18 98 269
Initialization of schemes for second-order equations      197
Integrability condition      312 370
Interior regularity estimate for finite difference schemes      330
Interior regularity estimate for partial differential equations      315
Interpolation operator      236
Irreducible matrix      349
Iteration matrix      341
Iteration matrix for      345
Jacobi method      340
Jacobi method for diagonally dominate matrices      349
Jacobi method, analysis of      345 346
Kreiss matrix theorem      225—233
Laplacc’s equation      311
Laplace transform      276 291
Laplace transform of a discrete function      277
Laplacian operator      311
Lax — Friedrichs scheme      17
Lax — Friedrichs scheme, stability      51
Lax — Richtmyer equivalence theorem      32—33
Lax — Richtmyer equivalence theorem for second-order equations      194
Lax — Richtmyer equivalence theorem, proof      262—266
Lax — Wendroff schcme      61 70
Lax — Wendroff schcme for parabolic equations      162
Lax — Wendroff schcme, dispersion      126
Lax — Wendroff schcme, dissipation      122
Lax — Wendroff schcme, modified      84
Lax — Wendroff schcme, smallest stencil      72
Lax — Wendroff schcme, stability of      76
Lcbesgue dominated convergence theorem      264 416
Leapfrog scheme for adding dissipation to      123
Leapfrog scheme for dispersion of      127
Leapfrog scheme for heat equation      147
Leapfrog scheme for hyperbolic equations      17 195 267
Leapfrog scheme for hyperbolic equations, explicit      100
Leapfrog scheme for hyperbolic equations, implicit      101
Leapfrog scheme for initialization      98
Leapfrog scheme for parasitic mode      99
Leapfrog scheme for stability of      95—97
Lebesgue integration      415
Lebesgue measure      414
Lexicographic order      340
Line Jacobi method      359
Linear iterative methods      341
Lower order terms and stability      53 149
Lower order terms and well-posedness of systems      218
MacCormack scheme      77
MacCormack scheme, time split      171
Mairix method for analyzing stability      307
Matrix norms      400
Matrix norms, formulas for      404
Maximum principle for analytic functions      422
Maximum principle for elliptic equations      317
Maximum principle for the discrete five-point Laplacian      326
Measurable function      414
Measurable set      414
Mitchell — Fairweather scheme      180
Monotone schemes      73
Multistep schemes      18 30 95
Multistep schemes as systems      167
Multistep schemes, convergence      24 267—269
Multistep schemes, dispersion of      127
Multistep schemes, initialization and order of accuracy      269
Neumann boundary condition      145 152 199 312 365—370
Norms for discrete functions      29
Norms for vectors      399
Norms in the factor space      367
Numerical boundary condition      85 281—288
One-way wave equation      1
Order of accuracy and initialization of multistep schemes      269
Order of accuracy and smoothness of parabolic equations      149
Order of accuracy for homogeneous equations      69
Order of accuracy for multistep schemes      267
Order of accuracy of a scheme      64
Order of accuracy of the solution      73
Order of accuracy, lower-order terms and stability of      149
Order of accuracy, parabolic equations      137
Order of accuracy, schemes for      145
Order of accuracy, using symbols      66
Parabolic systems      143 216
Parasitic mode      99
Parasitic mode, dispersion      128
Parseval’s relations      39
Peaceman — Rachford algorithm      175 181
Periodic problems      14
Periodic tridiagonal systems      91
Phase error      126 203
Phase error for multistep schemes      127
Poisson summation formula      250
Poisson’s equation      311
Polar coordinates      333
Positive definite matrix for elliptic schemes      336
Positive definite, iterative method for      362
Preconditioned conjugate gradient method      390
Quasi-characteristic extrapolation      86 282 292 294
Rayleigh equation      191 200
Reducible matrix      349
Reentrant corners      324 331
Regularity estimates      314 315 330
Residual      374
Resolvent condition for finite difference equations      227 289
Resolvent condition for partial differential equations      301
Restricted stability condition      50
Reverse Lax — Friedrichs scheme      36 58
Riemann integral      416
Robin condition      322
Robustness      206
Rouchfi’s theorem      110
Scalar product      401
SchrOdinger equation.      191
Schur polynomial      108 109 125 198
Schur’s lemma      403
Schwartz class      46
Search direction      378
Second-order equations      187
Second-order equations, convergence estimates for      270
Semi simple eigenvalue      404
Sherman — Morrison formula      91
Simple root      104
Simultaneously diagonalizable      169
SOR      340
SOR and Neumann boundary condition      368
SOR, analysis of      351
SOR, efficiency of      356
SOR, estimating the parameter      358
SOR, implementation      360
SOR, line      359
Spectral radius      341 404
SSOR      359 391
SSOR, preconditioner      391
Stability and lower-order terms      53
Stability and variable coefficients      59
Stability and von Neumann polynomial      108
Stability for ADI methods      177
Stability for ADI methods, condition, general      50
Stability for ADI methods, definition      28
Stability for initial-boundary value problems      288
Stability for multistep schemes      105
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