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数学中国—罂粟

原版英文书 第二版
! G) s' T! B) N& S2 [contents:
Preface to the first edition page viii
Preface to the second edition xi
5 F/ p6 v7 x  [7 T6 W- w) q; J1 Introduction 1
/ L  G2 A1 W5 {% k2 Parabolic equations in one space variable 7
2.1 Introduction 7
% j- e0 r$ e" N/ D2.2 A model problem 7
5 T6 T# |$ N7 }$ a" X2.3 Series approximation 9
2.4 An explicit scheme for the model problem 10
% ~$ U& K; Y" G1 ?8 r2.5 Difference notation and truncation error 12
7 H# P7 \; I9 C) ~  Y5 i2.6 Convergence of the explicit scheme 16
2.7 Fourier analysis of the error 19
/ b8 @5 g& b' j8 S6 N2.8 An implicit method 22
2.9 The Thomas algorithm 24
2.10 The weighted average or θ-method 26
' U; \% Z) g/ ?5 k2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
' d! _1 u( Y, ~6 P2 33
/ m3 n8 Z5 z* A& ~8 U( u7 G8 L2.12 A three-time-level scheme 38
2.13 More general boundary conditions 39
* ^: N5 ^, J6 E2.14 Heat conservation properties 44
9 E, {( j$ A9 x; E, E& y2.15 More general linear problems 46
: w  S9 a" u5 Y7 E! F2.16 Polar co-ordinates 52
3 {4 a1 y& z7 L2.17 Nonlinear problems 54
+ E2 i8 N* ]' C/ U% n: |8 n% ]Bibliographic notes 56
Exercises 56
! S* ?0 @4 Y- |v
vi Contents
3 2-D and 3-D parabolic equations 62
3.1 The explicit method in a rectilinear box 62
% K% Y* n6 j' {. g3.2 An ADI method in two dimensions 64
3.3 ADI and LOD methods in three dimensions 70
" W  @% H" H  S0 @6 u1 t4 X' j. I$ n3.4 Curved boundaries 71
3.5 Application to general parabolic problems 80
- \% @: V+ G% wBibliographic notes 83
Exercises 83
2 o8 O# K0 E9 J- M* K& L4 Hyperbolic equations in one space dimension 86
" m% _! M# y% Z" \1 l1 z; C+ r4.1 Characteristics 86
4.2 The CFL condition 89
* r7 E* [0 {! c0 p# A1 {) j4.3 Error analysis of the upwind scheme 94
3 g1 g. r$ d2 ^5 Z' }7 c1 T4.4 Fourier analysis of the upwind scheme 97
4.5 The Lax–Wendroff scheme 100
/ a. i1 r0 C+ f: B. l4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
* _: V) ~- o1 n9 e' y4.8 The box scheme 116
4.9 The leap-frog scheme 123
4.10 Hamiltonian systems and symplectic
integration schemes 128
5 E) [9 U+ X' I7 z8 V4.11 Comparison of phase and amplitude errors 135
2 k. x1 ^8 i1 K( i7 Z) E. r4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
Bibliographic notes 146
4 p; C1 J% ~# _# VExercises 146
$ K+ F$ s& U1 N" I# j! B5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
# b' ~# ^* i; X5 g. U9 x8 r% X2 q1 o5.2 The finite difference mesh and norms 152
, M: x9 A8 I7 u# T1 E# e. T' |( t5.3 Finite difference approximations 154
  }% z& C- o: |1 W* t( \5 E. e5.4 Consistency, order of accuracy and convergence 156
5.5 Stability and the Lax Equivalence Theorem 157
4 \* g% r9 Y. _) l4 m7 [7 j5.6 Calculating stability conditions 160
% c& g1 a8 z& V9 D$ o5.7 Practical (strict or strong) stability 166
4 }; _% a4 I6 D$ t! p5.8 Modified equation analysis 169
" E: Z' n8 c0 p2 Y5.9 Conservation laws and the energy method of analysis 177
( o6 q' C) C3 @. n5.10 Summary of the theory 186
Bibliographic notes 189
Exercises 190
Contents vii
! {' T# m3 ^# N% y5 p' W" f6 Linear second order elliptic equations in
two dimensions 194
7 V. I% c' i  }) ]2 P6.1 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
6 C3 W) s' H; j- p' I* t6.5 Error analysis using a maximum principle 203
& N0 ]5 e% y. I+ X7 g5 W* t6.6 Asymptotic error estimates 213
, m2 V! q. i( C$ p6.7 Variational formulation and the finite
element method 218
6.8 Convection–diffusion problems 224
6.9 An example 228
7 u9 g/ O8 R: \Bibliographic notes 231
Exercises 232
5 F; N( V, L0 M$ Z4 a1 W  F0 f7 Iterative solution of linear algebraic equations 235
7.1 Basic iterative schemes in explicit form 237
7.2 Matrix form of iteration methods and
; o* y, P1 f0 u7 e9 }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
7.6 The multigrid method 252
" B  a" e5 |1 ~4 [7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
Bibliographic notes 263
$ g3 A! W! {, Z+ RExercises 263
6 f0 X& q& J' D  ]* tReferences 267
Index 273

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