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数学中国—罂粟

原版英文书 第二版
contents:
Preface to the first edition page viii
- L1 Y% b( h. N/ J( MPreface to the second edition xi
1 Introduction 1
- R/ I( [! d$ L( l: n' E, q8 @2 G2 Parabolic equations in one space variable 7
  a8 ^5 u; I) t' e! r2.1 Introduction 7
2.2 A model problem 7
7 F8 D! Q6 O& t: q2.3 Series approximation 9
2.4 An explicit scheme for the model problem 10
2.5 Difference notation and truncation error 12
4 {0 Q! Q% J6 l9 R$ H* `2.6 Convergence of the explicit scheme 16
9 ~3 n3 J7 V7 a5 p$ s2.7 Fourier analysis of the error 19
2.8 An implicit method 22
. B  B. M% ^4 r# u: z8 o) l6 E2.9 The Thomas algorithm 24
2.10 The weighted average or θ-method 26
# a* e- W1 Y0 F8 ]6 D2.11 A maximum principle and convergence
' I! m5 o: r' ~  `' r. T, Gfor μ(1−θ)≤ 1
1 w" G$ x1 _4 B2 33
2.12 A three-time-level scheme 38
6 f1 o' b, n& @* T2.13 More general boundary conditions 39
2 W4 u) G! W+ K6 H- c, p. {2.14 Heat conservation properties 44
) j8 {! ]$ i4 \- V- t2.15 More general linear problems 46
' }" O& i7 B  @) ^3 R! G; Y2.16 Polar co-ordinates 52
2.17 Nonlinear problems 54
, D$ U/ W0 M/ h# d/ I0 oBibliographic notes 56
" v+ w- B+ u3 s& q8 r. @4 _6 k: DExercises 56
8 r/ i) q& j9 K2 J6 G0 ^v
$ o- u) a  V! D" r& W0 _6 L2 Z# Nvi Contents
/ ?7 Y6 Z8 ^9 u1 s0 {% I. b3 \3 2-D and 3-D parabolic equations 62
7 T- s! d- C/ C; f: w# r/ l0 C3.1 The explicit method in a rectilinear box 62
0 |( T7 l- @3 j7 N3 H' ^' F3.2 An ADI method in two dimensions 64
3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
3.5 Application to general parabolic problems 80
Bibliographic notes 83
7 l: |6 }, {/ ^% q+ B: P  \# MExercises 83
4 Hyperbolic equations in one space dimension 86
4.1 Characteristics 86
0 K2 ]4 [$ V, d: w. O, E4.2 The CFL condition 89
* f  t: v& @  n4.3 Error analysis of the upwind scheme 94
4.4 Fourier analysis of the upwind scheme 97
. X# {  h8 X, V' B: a4.5 The Lax–Wendroff scheme 100
3 i5 ^# @. T* @7 v6 o; }4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
4.8 The box scheme 116
/ e7 G5 o2 F& U  V! }7 ~& u" F4.9 The leap-frog scheme 123
0 R3 b' Q- I+ ]0 A( p0 M4.10 Hamiltonian systems and symplectic
1 X, `. \, k. v( t" x: ]integration schemes 128
9 Y% E# [1 _6 Q5 d3 S  C6 L4.11 Comparison of phase and amplitude errors 135
4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
; v, ^% h3 ~# b* N+ cBibliographic notes 146
Exercises 146
+ Q. h9 n! n8 ^7 q) z; `5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
$ F8 F, W0 d) I- e2 T# a; I5.2 The finite difference mesh and norms 152
5.3 Finite difference approximations 154
2 m3 C4 J: J4 u7 d9 e5.4 Consistency, order of accuracy and convergence 156
' n9 H7 ]4 c! n2 g. X0 u5.5 Stability and the Lax Equivalence Theorem 157
5.6 Calculating stability conditions 160
, z4 y/ M5 y0 ^- g" A5.7 Practical (strict or strong) stability 166
/ m6 |" [4 `. Z1 w: i# A4 d5.8 Modified equation analysis 169
; c) q9 }" @) ?# B( s% {5.9 Conservation laws and the energy method of analysis 177
: |9 e) c) \4 f1 k7 b: C: O4 O5.10 Summary of the theory 186
! c9 a- g3 o- `) Q8 X1 ]Bibliographic notes 189
Exercises 190
Contents vii
6 Linear second order elliptic equations in
two dimensions 194
6.1 A model problem 194
& C5 y; Z7 p" J- f- `1 {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
. M3 |$ W2 Z2 P: B1 n8 G3 H6.5 Error analysis using a maximum principle 203
6.6 Asymptotic error estimates 213
6.7 Variational formulation and the finite
element method 218
: G# L# G0 i- h+ D; U6.8 Convection–diffusion problems 224
6.9 An example 228
Bibliographic notes 231
Exercises 232
6 u* Z% u" F  k1 X8 b7 Iterative solution of linear algebraic equations 235
7.1 Basic iterative schemes in explicit form 237
9 |3 t# R$ j; n; F6 E% |' I7.2 Matrix form of iteration methods and
their convergence 239
; [8 H- U: i! I$ q. @7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
3 D/ u* \' Q# ?' w  l- L+ Z7.6 The multigrid method 252
7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
Bibliographic notes 263
$ m; D. J  z8 W+ G% p' \) HExercises 263
5 W- p8 f5 G7 Y1 M7 d- dReferences 267
Index 273

' }* y' W" m; r( z( v  @0 P& e

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