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
contents:
1 y( C+ w6 ]# XPreface to the first edition page viii
( [5 W  @+ }% n8 g, i( P9 p+ x9 CPreface to the second edition xi
. ^% {1 U8 t; ^/ T1 Introduction 1
; m& d. {1 S! V- P4 s$ @, R. g2 Parabolic equations in one space variable 7
9 r* h1 V4 z' \2.1 Introduction 7
2.2 A model problem 7
. q6 T( G# s1 N5 v( K7 ]9 S, G2.3 Series approximation 9
2 t) I! ~) o3 v- b2.4 An explicit scheme for the model problem 10
& c' k( R4 A7 @2 |2.5 Difference notation and truncation error 12
2.6 Convergence of the explicit scheme 16
* d5 a3 W: i2 E4 L2.7 Fourier analysis of the error 19
% Z# b8 f4 Q" s5 ?2.8 An implicit method 22
7 j4 G; b5 X6 O2 v2 d1 c# k2.9 The Thomas algorithm 24
2.10 The weighted average or θ-method 26
2.11 A maximum principle and convergence
6 X4 d1 }+ B2 P# f) O4 Gfor μ(1−θ)≤ 1
2 33
! M/ y7 u; p5 N( s, i$ y) ]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
6 X2 I& i) a' \# L2.16 Polar co-ordinates 52
+ ~+ `" V" j& q1 Y. `1 E2.17 Nonlinear problems 54
Bibliographic notes 56
. L5 o8 K) L  v; ]# d4 ]) w, qExercises 56
v
/ d  [( B- h# n* Hvi Contents
3 2-D and 3-D parabolic equations 62
. i9 z$ i6 {2 W. p- L! D* H+ `3.1 The explicit method in a rectilinear box 62
3.2 An ADI method in two dimensions 64
$ W8 ]3 Q. Q3 l( `/ U3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
3.5 Application to general parabolic problems 80
Bibliographic notes 83
Exercises 83
# U- q' e3 u" ?4 l/ f4 Hyperbolic equations in one space dimension 86
! i& ~1 m# e! [, M8 o" L+ N# q/ S4.1 Characteristics 86
% v' Y# a' T0 ~! E4.2 The CFL condition 89
# h, f8 R' w0 i4 j4.3 Error analysis of the upwind scheme 94
4.4 Fourier analysis of the upwind scheme 97
  N" }5 l, v( R5 `/ o4.5 The Lax–Wendroff scheme 100
4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
' _" R6 m( Z9 n7 Z3 B" q4.8 The box scheme 116
3 m+ [/ o2 z$ T4.9 The leap-frog scheme 123
/ _" D, x- N& g2 E. D4.10 Hamiltonian systems and symplectic
! h: R, w* E: r) f4 yintegration schemes 128
4.11 Comparison of phase and amplitude errors 135
- R" M1 }4 w; Y/ E% c, q4.12 Boundary conditions and conservation properties 139
4.13 Extensions to more space dimensions 143
Bibliographic notes 146
Exercises 146
$ C2 y& o6 H  v" `: p' D5 B7 t* C* n5 Consistency, convergence and stability 151
7 E% [/ V2 Q2 ?5.1 Definition of the problems considered 151
, D* N0 _" W; u7 G; B5.2 The finite difference mesh and norms 152
: H+ U/ J& W  w1 Y! E5.3 Finite difference approximations 154
5.4 Consistency, order of accuracy and convergence 156
5.5 Stability and the Lax Equivalence Theorem 157
; v& U' o' o$ I$ W; b. U# i5.6 Calculating stability conditions 160
5.7 Practical (strict or strong) stability 166
5.8 Modified equation analysis 169
2 N& X* E# K& V' k" [- U' r5.9 Conservation laws and the energy method of analysis 177
3 D) E: w( X; P( q4 r/ I* J7 o% \5.10 Summary of the theory 186
Bibliographic notes 189
6 y4 P$ Z& B! bExercises 190
Contents vii
6 Linear second order elliptic equations in
two dimensions 194
7 B6 E2 ^; f2 s) U) Y4 \1 D- r$ @6.1 A model problem 194
# U0 \8 l- m! X; n; g( Y6.2 Error analysis of the model problem 195
/ K. N" u8 ^  E$ V7 o2 _6 u6.3 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
8 X. |0 k/ F; U! B' l, x6.7 Variational formulation and the finite
element method 218
  ]+ ~2 C* E1 t% `; P6.8 Convection–diffusion problems 224
6.9 An example 228
Bibliographic notes 231
Exercises 232
7 Iterative solution of linear algebraic equations 235
6 c/ W3 M8 R+ N7.1 Basic iterative schemes in explicit form 237
7.2 Matrix form of iteration methods and
' E& A+ P, e) F) e  Ytheir convergence 239
. _8 e4 b1 `* |$ ~/ \7.3 Fourier analysis of convergence 244
7 e1 Y: q# Y7 {7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
: k  G# M& K! p% R7.6 The multigrid method 252
( U  `% w2 b& W9 A8 k; L7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
  m. D! B% R, S. N4 |Bibliographic notes 263
Exercises 263
References 267
Index 273

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