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
( D5 d. d+ v* S. n' w2 J& Ocontents:
7 f& z: `* L/ b+ X5 e$ a- D9 I$ HPreface to the first edition page viii
# T6 p1 h4 @3 a9 pPreface to the second edition xi
# K$ f, `, l2 v0 S9 B7 ~' U1 Introduction 1
7 A$ H  |  N& p1 S8 ?5 i2 Parabolic equations in one space variable 7
2.1 Introduction 7
" [; O$ f8 M# L& a( v2.2 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
" ~( P" G7 C4 B+ l! R9 ?$ J9 s2 R1 X2.6 Convergence of the explicit scheme 16
  h/ O8 ~. I+ g% R* ^# v( Y5 O2.7 Fourier analysis of the error 19
* r# c( O/ }3 n0 ~8 |9 W! r' \2 C: w2.8 An implicit method 22
) T( ^. V" D! j, y( J# Q2.9 The Thomas algorithm 24
+ e, u2 Q4 j1 q3 w2.10 The weighted average or θ-method 26
( L# T: n1 V( Z4 w! b6 B2.11 A maximum principle and convergence
9 c% V: a0 I* F! K) xfor μ(1−θ)≤ 1
2 33
2.12 A three-time-level scheme 38
2.13 More general boundary conditions 39
2.14 Heat conservation properties 44
2.15 More general linear problems 46
4 J7 Z0 c- z% {; h. q2.16 Polar co-ordinates 52
2.17 Nonlinear problems 54
Bibliographic notes 56
7 }; R# z2 E: R2 s; MExercises 56
v
5 t& |4 m0 D1 P  N: w+ I8 W0 J6 fvi Contents
7 @& S5 T( N6 S0 }( B3 2-D and 3-D parabolic equations 62
6 z/ {  T* L( z5 N& k3.1 The explicit method in a rectilinear box 62
5 V6 e- K+ U1 n' L8 w3.2 An ADI method in two dimensions 64
3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
9 h, Y: B$ J. [) g  n3.5 Application to general parabolic problems 80
Bibliographic notes 83
Exercises 83
3 a  m4 F* {9 H' C) k/ |. f3 H9 m4 Hyperbolic equations in one space dimension 86
4.1 Characteristics 86
0 z' `$ `7 x; M. ]  m8 ?4.2 The CFL condition 89
4.3 Error analysis of the upwind scheme 94
8 B& h  M& m9 g0 Y0 T4 _0 E4.4 Fourier analysis of the upwind scheme 97
4.5 The Lax–Wendroff scheme 100
4.6 The Lax–Wendroff method for conservation laws 103
5 K; N; \% S% F( u6 X4.7 Finite volume schemes 110
+ B8 \: L2 w+ \4.8 The box scheme 116
6 @! g2 k7 W, V( N2 y  v4.9 The leap-frog scheme 123
7 _' U; B( K- c# `6 W( M. a0 x4.10 Hamiltonian systems and symplectic
integration schemes 128
4.11 Comparison of phase and amplitude errors 135
& v, ]  i4 W& x" Y9 H+ O5 q4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
& |) L1 w, W  j8 eBibliographic notes 146
Exercises 146
5 Consistency, convergence and stability 151
& P7 v8 z2 C3 ~4 R6 i% x& w5.1 Definition of the problems considered 151
( b  }( z# q) ]4 ~" L4 p5.2 The finite difference mesh and norms 152
5.3 Finite difference approximations 154
# l& T' F7 G4 b3 j/ X/ V& R5.4 Consistency, order of accuracy and convergence 156
5.5 Stability and the Lax Equivalence Theorem 157
5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
3 ]2 H: l% H0 |4 w' c# S) u5.8 Modified equation analysis 169
5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
5 u' {# B# ]  [Bibliographic notes 189
3 Z% ?) E5 H0 m7 ]  R: kExercises 190
Contents vii
0 ^" d6 P2 z0 ]- Z6 Linear second order elliptic equations in
  u8 S8 p; w( Ztwo dimensions 194
/ p0 e' H0 T9 ?5 _9 I  R5 Y6 C6.1 A model problem 194
6.2 Error analysis of the model problem 195
6.3 The general diffusion equation 197
6.4 Boundary conditions on a curved boundary 199
% B% H0 i$ Q/ C  `6.5 Error analysis using a maximum principle 203
/ U6 B) @( Y, Q. q( t" ^6.6 Asymptotic error estimates 213
7 I6 E$ D! a4 {" O/ k/ f# s6 m4 F  L& {6.7 Variational formulation and the finite
" {# p' a7 t0 N3 Oelement method 218
6.8 Convection–diffusion problems 224
1 g/ V2 {5 Q1 r% Z, ^4 ~% X* N6.9 An example 228
. M7 z) O# p5 TBibliographic notes 231
; L+ g% f  r3 {9 EExercises 232
7 Iterative solution of linear algebraic equations 235
7.1 Basic iterative schemes in explicit form 237
7.2 Matrix form of iteration methods and
0 L$ B* z; c6 U0 V9 i$ m- ftheir convergence 239
1 D1 {" g3 O+ N$ I  y! h  W5 q3 [7.3 Fourier analysis of convergence 244
8 K" F1 ^1 K8 ]2 P7 t/ [) m7.4 Application to an example 248
  R( a- X0 u* y9 i& m7.5 Extensions and related iterative methods 250
7.6 The multigrid method 252
7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
Bibliographic notes 263
Exercises 263
  f1 N; n5 L# c4 W: IReferences 267
- M; X$ I- w! B+ q. a( P  |Index 273

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