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:
Preface to the first edition page viii
- p1 G6 ?& z. _5 w& IPreface to the second edition xi
2 k# @7 t: V2 R1 Introduction 1
! R/ u, l6 e2 x1 ?4 B2 Parabolic equations in one space variable 7
2.1 Introduction 7
2.2 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
  M0 W, n- ^% E$ N2.6 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
2.10 The weighted average or θ-method 26
2.11 A maximum principle and convergence
$ s9 V# v1 o# C+ V8 k% afor μ(1−θ)≤ 1
2 33
1 z5 z& ?! v  B8 r2.12 A three-time-level scheme 38
2.13 More general boundary conditions 39
% H# V) i! z3 x9 T( @' i2.14 Heat conservation properties 44
2.15 More general linear problems 46
2.16 Polar co-ordinates 52
' G* G* y- A* w- f2.17 Nonlinear problems 54
Bibliographic notes 56
( q$ \( m6 y. o& n  @2 dExercises 56
+ @' ~6 j0 s  gv
vi Contents
; D8 Y) I% O; p7 Q4 l( V# b6 h3 2-D and 3-D parabolic equations 62
6 V, Y, ?1 L# m( `+ k/ Q: Q3.1 The explicit method in a rectilinear box 62
1 o( `* G0 i5 u( k1 o$ P3.2 An ADI method in two dimensions 64
3.3 ADI and LOD methods in three dimensions 70
3.4 Curved boundaries 71
) N# m- F) n" m" F, O# |3.5 Application to general parabolic problems 80
Bibliographic notes 83
Exercises 83
+ L* j" ]" Y: q3 }0 z9 k& P9 K* w4 Hyperbolic equations in one space dimension 86
4.1 Characteristics 86
4.2 The CFL condition 89
+ j/ H* ]9 m4 x( i. S4.3 Error analysis of the upwind scheme 94
) U+ A' ^* i0 T# M: b- F4.4 Fourier analysis of the upwind scheme 97
) w9 V% s: S0 ?1 g4.5 The Lax–Wendroff scheme 100
4.6 The Lax–Wendroff method for conservation laws 103
4.7 Finite volume schemes 110
8 D# `; [0 \. _7 |4.8 The box scheme 116
& n+ w4 K' X$ v! z2 r8 \" h! N' |4.9 The leap-frog scheme 123
3 ~1 D! M; |: f0 c; _0 u4.10 Hamiltonian systems and symplectic
  r: [% k& j0 o9 w, J  l5 T2 f- hintegration schemes 128
4.11 Comparison of phase and amplitude errors 135
4.12 Boundary conditions and conservation properties 139
8 F4 b7 ~: J# @( j: d% b, p4.13 Extensions to more space dimensions 143
Bibliographic notes 146
Exercises 146
) b6 |( e5 e4 I5 Consistency, convergence and stability 151
5.1 Definition of the problems considered 151
5.2 The finite difference mesh and norms 152
6 ]2 f5 ~" {. `5.3 Finite difference approximations 154
9 c0 Z9 ^6 \' l5.4 Consistency, order of accuracy and convergence 156
$ Q0 ?5 d. W: f% f7 v0 w/ Z" \- y5.5 Stability and the Lax Equivalence Theorem 157
5.6 Calculating stability conditions 160
1 n- s  Z3 b% S2 P% i* Z5.7 Practical (strict or strong) stability 166
5.8 Modified equation analysis 169
$ _" H6 a: y9 D7 Z+ [% O5.9 Conservation laws and the energy method of analysis 177
, J; c9 [+ ^8 s0 D5.10 Summary of the theory 186
* u$ W; }: {3 `8 P: M+ v5 ABibliographic notes 189
1 f$ R* o& }: [) t+ w8 }6 u* n7 g! \Exercises 190
) b2 n# _' j: S  BContents vii
) {' \- g. V7 K6 Linear second order elliptic equations in
, F( u1 S) e* U& n( {two dimensions 194
0 `- j+ E* G( ?1 H6.1 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
6.5 Error analysis using a maximum principle 203
6.6 Asymptotic error estimates 213
+ A  `" ^3 J3 G0 A6.7 Variational formulation and the finite
element method 218
6.8 Convection–diffusion problems 224
9 W( ^/ p2 v6 {* [4 w) P- [6.9 An example 228
Bibliographic notes 231
& k; k; m& _3 d* j4 f" t' ^Exercises 232
7 Iterative solution of linear algebraic equations 235
7.1 Basic iterative schemes in explicit form 237
* C7 X6 W6 t7 L2 u: e$ k7.2 Matrix form of iteration methods and
& a' t4 \+ r, P/ Utheir convergence 239
' j1 z( O, @- I% f* |5 p0 F5 N& ^7.3 Fourier analysis of convergence 244
: s+ f/ Z, ~- K5 \5 |% q7.4 Application to an example 248
7.5 Extensions and related iterative methods 250
7.6 The multigrid method 252
9 P+ h( E, r* K! W4 |, S- G9 S1 t7 f7.7 The conjugate gradient method 258
7.8 A numerical example: comparisons 261
Bibliographic notes 263
: k/ S" u$ S" ~$ W* U  cExercises 263
References 267
& v( A! |/ k. ~Index 273

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