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数学中国—罂粟

原版英文书 第二版
4 k2 S, N; m8 a& m( g$ t9 I- Fcontents:
Preface to the first edition page viii
Preface to the second edition xi
  ^& K# F3 j; X3 s" [7 V1 r" @9 p3 _1 Introduction 1
" F6 w# k, t' V: d2 }- ~2 Parabolic equations in one space variable 7
- v8 e  ^. v) A; j% i# I8 R! w) o! n3 v2.1 Introduction 7
2.2 A model problem 7
2.3 Series approximation 9
8 g& M* D9 T) e1 Z2.4 An explicit scheme for the model problem 10
0 `2 ]# U4 ^! c7 P% E5 `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 K, D- a& |1 |0 |. h2.10 The weighted average or θ-method 26
2.11 A maximum principle and convergence
for μ(1−θ)≤ 1
/ ^; z5 |# v' D) m2 U; `2 33
2.12 A three-time-level scheme 38
2.13 More general boundary conditions 39
- R) w2 D$ @& x/ f3 j7 H4 \% B2.14 Heat conservation properties 44
2.15 More general linear problems 46
2.16 Polar co-ordinates 52
. o; Q! f7 G$ `6 F1 l2.17 Nonlinear problems 54
Bibliographic notes 56
Exercises 56
: G; P  ?* H- [/ @v
8 w$ J8 L& o3 M" V( Y/ nvi Contents
! O- j# h# O. X; G3 2-D and 3-D parabolic equations 62
3.1 The explicit method in a rectilinear box 62
& `5 L/ }2 ?; \0 B4 f7 d3.2 An ADI method in two dimensions 64
' N1 f& w0 M% Z' M3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
3.5 Application to general parabolic problems 80
Bibliographic notes 83
* t& j" o7 O6 F% g$ p8 EExercises 83
4 Hyperbolic equations in one space dimension 86
. L& {' K( r( B$ m4.1 Characteristics 86
4.2 The CFL condition 89
, o% x( t/ q; u4.3 Error analysis of the upwind scheme 94
' N$ Y- L5 \+ U) D4.4 Fourier analysis of the upwind scheme 97
4 I+ ?$ s7 O( B6 j. y0 k6 T4 h4.5 The Lax–Wendroff scheme 100
# f& k4 v  p# j4.6 The Lax–Wendroff method for conservation laws 103
6 X5 F1 Q# \/ U; S1 p4.7 Finite volume schemes 110
4.8 The box scheme 116
- t" y9 K6 I6 B# h' \9 U' t4 N2 B  y0 \% G4.9 The leap-frog scheme 123
4.10 Hamiltonian systems and symplectic
' x3 T$ `+ b+ A6 {, L. b5 zintegration schemes 128
) w0 ]0 E" s8 O4.11 Comparison of phase and amplitude errors 135
4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
Bibliographic notes 146
Exercises 146
. Z- ?' i6 R9 i: y5 Consistency, convergence and stability 151
+ W, _% c" o3 W* c5.1 Definition of the problems considered 151
  u2 Y0 d, ]' f* H4 H5.2 The finite difference mesh and norms 152
5.3 Finite difference approximations 154
5.4 Consistency, order of accuracy and convergence 156
5 x1 {: y5 `5 Q  A6 c5.5 Stability and the Lax Equivalence Theorem 157
& |4 J  {! ]. o: Y1 s7 @, X5.6 Calculating stability conditions 160
0 t7 \& |$ }3 [0 S9 e5.7 Practical (strict or strong) stability 166
: J+ Y3 q, ^4 P/ S2 s: \5.8 Modified equation analysis 169
) M- p1 S9 M) J5.9 Conservation laws and the energy method of analysis 177
5.10 Summary of the theory 186
Bibliographic notes 189
Exercises 190
+ H. f( F8 Z5 I' BContents vii
# B) F* z& G2 \& ^6 u( z6 Linear second order elliptic equations in
two dimensions 194
; R8 I$ p; [# F6.1 A model problem 194
6.2 Error analysis of the model problem 195
6.3 The general diffusion equation 197
3 l7 r& t+ H. F/ C6.4 Boundary conditions on a curved boundary 199
4 w! O6 T; Z/ C9 i/ @9 D" c% a6.5 Error analysis using a maximum principle 203
9 |; b% E+ {" `% N. W* y6.6 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
Bibliographic notes 231
Exercises 232
- Q" ?' k" y, p+ E8 S* v7 Iterative solution of linear algebraic equations 235
- H; l# j* @; T- T! K, B7.1 Basic iterative schemes in explicit form 237
7.2 Matrix form of iteration methods and
their convergence 239
+ a" r5 D0 w% p$ [7 O- \6 g2 |7.3 Fourier analysis of convergence 244
7.4 Application to an example 248
. r1 X* }5 T% u, Z2 ?7.5 Extensions and related iterative methods 250
3 x, |& R: D$ Y" ~8 W7.6 The multigrid method 252
7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
. L6 d* {3 X. g) z* P% c- LBibliographic notes 263
& P1 C& L2 l, F0 R& }Exercises 263
References 267
8 ?( D0 G/ I, V8 G0 ?! oIndex 273

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