剑桥出版社---偏微分方程数值解法

数学中国—罂粟

原版英文书 第二版
- j6 `7 `0 l* |3 t& t& o' @contents:
Preface to the first edition page viii
Preface to the second edition xi
7 ?9 \, V8 I7 W* V- `$ K. _$ {! D1 Introduction 1
2 Parabolic equations in one space variable 7
8 R; L' E# w9 ]+ i2.1 Introduction 7
2.2 A model problem 7
2.3 Series approximation 9
2.4 An explicit scheme for the model problem 10
2.5 Difference notation and truncation error 12
2.6 Convergence of the explicit scheme 16
2.7 Fourier analysis of the error 19
2.8 An implicit method 22
2.9 The Thomas algorithm 24
2.10 The weighted average or θ-method 26
/ c8 Z! D- z3 r0 U( a" p6 _2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
2 33
* ]$ t; @$ d; K' p5 h, h2.12 A three-time-level scheme 38
9 s* D: U: g8 |- e* [4 f6 D2 L2.13 More general boundary conditions 39
2.14 Heat conservation properties 44
2.15 More general linear problems 46
$ B  T" p% x$ u% Q& l) F2.16 Polar co-ordinates 52
2.17 Nonlinear problems 54
Bibliographic notes 56
Exercises 56
) F! ~3 t8 }# }* _8 H2 n. [6 J0 Sv
( I4 d8 A1 P& E! X! h8 Mvi Contents
3 |1 ~1 w- W, v  I% L1 k3 2-D and 3-D parabolic equations 62
3.1 The explicit method in a rectilinear box 62
* ~: G: R' r/ i. q7 G# v3.2 An ADI method in two dimensions 64
3.3 ADI and LOD methods in three dimensions 70
" w9 `+ O* B* v% |" e3.4 Curved boundaries 71
" D/ l. n3 {% H9 b, g3.5 Application to general parabolic problems 80
, \: K$ O( U) l2 T% P+ n( bBibliographic notes 83
2 d. o! K# q2 H; ~( OExercises 83
4 Hyperbolic equations in one space dimension 86
4.1 Characteristics 86
4.2 The CFL condition 89
4.3 Error analysis of the upwind scheme 94
" M8 m' ]1 F7 g% h# w4.4 Fourier analysis of the upwind scheme 97
4.5 The Lax–Wendroff scheme 100
8 g. N$ ]& Y% Q8 h$ \* ^4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
  i* t/ z5 g- w6 v; C4.8 The box scheme 116
4.9 The leap-frog scheme 123
. D% q) ?+ {8 z6 _4.10 Hamiltonian systems and symplectic
integration schemes 128
8 N! P9 T1 s" e1 m2 x/ c) P4.11 Comparison of phase and amplitude errors 135
: ^/ e) m& j& H/ E1 p4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
/ L+ Y9 x. k$ QBibliographic notes 146
Exercises 146
5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
  b, m/ b( ^) v8 M# Y4 R5.2 The finite difference mesh and norms 152
5.3 Finite difference approximations 154
5.4 Consistency, order of accuracy and convergence 156
: s' f5 _2 d. o$ ?" k5.5 Stability and the Lax Equivalence Theorem 157
5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
5.8 Modified equation analysis 169
4 P, R9 `0 g$ m$ f* O+ A7 T$ v5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
Bibliographic notes 189
& U0 k! g* n( q0 R) OExercises 190
Contents vii
% _) V7 l' f5 q7 D: p) l6 h6 Linear second order elliptic equations in
two dimensions 194
! X2 J8 k/ h9 N# a0 h2 m6.1 A model problem 194
6.2 Error analysis of the model problem 195
$ L$ {4 x# f9 b6.3 The general diffusion equation 197
6.4 Boundary conditions on a curved boundary 199
6.5 Error analysis using a maximum principle 203
6.6 Asymptotic error estimates 213
6.7 Variational formulation and the finite
element method 218
6.8 Convection–diffusion problems 224
6.9 An example 228
! r5 v% R/ T' K% uBibliographic notes 231
Exercises 232
- a2 q  i4 Q; V% V3 F$ L5 w7 Iterative solution of linear algebraic equations 235
7.1 Basic iterative schemes in explicit form 237
8 S6 n3 h+ R& d# N: t' A: g' D% _7.2 Matrix form of iteration methods and
their convergence 239
7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
# J3 y+ b4 g; ^3 f' I# m: W7.6 The multigrid method 252
7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
Bibliographic notes 263
1 `3 w6 s2 V6 Y0 g# x2 ^Exercises 263
References 267
' P4 P( i& y; z0 j6 RIndex 273

* ]8 b9 ]$ v$ Q1 Y. }

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