剑桥出版社---偏微分方程数值解法 - 数学建模社区-数学中国

原版英文书第二版
  L0 L$ ?3 a( d$ z* C- Y+ [contents:
6 a  i* D: V! B7 HPreface to the first edition page viii
  A2 Y) p  }5 SPreface to the second edition xi
5 Z6 E  O  t' J1 Introduction 1* O" i2 {- K( H
2 Parabolic equations in one space variable 7# A7 r0 p3 ^7 K% ?2 u- Y- p
2.1 Introduction 7
9 q/ Z0 u0 y3 m: C! P! V, g2.2 A model problem 74 q7 x/ `" l# L  u* i* ~
2.3 Series approximation 9
8 H! |4 C4 `7 R# Y2.4 An explicit scheme for the model problem 100 _2 D3 F3 m. ?! _, c4 J; S/ L! O
2.5 Difference notation and truncation error 12
2 b7 K' Y4 E. V7 Q' K" n2.6 Convergence of the explicit scheme 16. X& [# h' L7 e* g6 |$ c
2.7 Fourier analysis of the error 19
* |7 y; u$ I3 o# ^4 U( E2.8 An implicit method 22& f& e; ?3 D: n6 W
2.9 The Thomas algorithm 24# `/ r* }# g2 a7 ?% N0 s$ X" [$ l
2.10 The weighted average or θ-method 26
3 ~* o, x9 ~0 D7 R2.11 A maximum principle and convergence, r! v( ?* q. M9 A9 v
for μ(1−θ)≤ 1
/ w, x2 I! u$ x+ ?; B2 33
* x, t5 P8 ?5 k$ w5 f3 K1 {2.12 A three-time-level scheme 382 G/ X2 C1 n& o' H7 |
2.13 More general boundary conditions 392 n8 n- |' `5 F: K( H. g
2.14 Heat conservation properties 44
/ ~  z- I! {. c" g/ h, J7 ?# [2.15 More general linear problems 46( U: o$ C1 [8 S' t. Z* n$ ]
2.16 Polar co-ordinates 52" A/ h$ ~4 I. ^
2.17 Nonlinear problems 545 y7 H6 j; }2 A. n5 \) v  ?
Bibliographic notes 56
3 s4 H9 B( W! {' m. {Exercises 56
5 W, e, ~7 o- Y+ uv
7 V/ s) S- Y/ V. b& Gvi Contents, L5 c6 E: b# d, Q6 A% t2 I
3 2-D and 3-D parabolic equations 62: E+ f, U# M/ i  s1 z+ J
3.1 The explicit method in a rectilinear box 62
* I5 R+ e! F8 x' B: H1 D3.2 An ADI method in two dimensions 64, \' w7 D" u9 p( c* S$ f
3.3 ADI and LOD methods in three dimensions 70$ o4 s. `3 {4 E* B9 `: M) _
3.4 Curved boundaries 71
; J! i; W. Y2 V" s9 F3.5 Application to general parabolic problems 80) Z, e0 ^9 Y  C4 p# u% p6 [/ A
Bibliographic notes 83
3 Z. T* i6 L4 t9 Z( rExercises 83
1 H. A% G" X, g/ L4 Hyperbolic equations in one space dimension 86" w0 P; O, G# v
4.1 Characteristics 861 M! Y2 S2 F2 M" s, z
4.2 The CFL condition 89- `" q$ p0 S) l" U$ W/ U. K6 C
4.3 Error analysis of the upwind scheme 94
! _4 t9 ^! Y* s$ R4.4 Fourier analysis of the upwind scheme 97# Q2 {7 o5 L) O/ Z6 x( G+ n
4.5 The Lax–Wendroff scheme 100
* g! u2 `1 s- D  }6 \0 N4.6 The Lax–Wendroff method for conservation laws 103
/ N( V7 b8 L) w( s4.7 Finite volume schemes 110
" O" _& [8 A" p4.8 The box scheme 116
4.9 The leap-frog scheme 123
% D/ u& M9 V7 P* ?  Z  }. D) ^9 W4.10 Hamiltonian systems and symplectic8 z( [9 j$ d. j" _7 l
integration schemes 128( F, B+ t5 X9 s" L$ ]7 X' J, K
4.11 Comparison of phase and amplitude errors 135
; G* B6 a3 _1 J, m8 `  u- u4.12 Boundary conditions and conservation properties 139
8 h- j9 s5 r- M* k- K% C3 n4.13 Extensions to more space dimensions 1432 W( K; C0 c  s9 `+ O" o: a
Bibliographic notes 146
3 G! D& n9 k5 w( b% R; P% \Exercises 146
. \" z/ |7 Z4 u5 Consistency, convergence and stability 151+ p2 f, z2 Y5 U! r6 I9 Q& P1 ]
5.1 Definition of the problems considered 151% f/ f- k9 U! G7 P3 a
5.2 The finite difference mesh and norms 1529 M# J2 k5 F3 y
5.3 Finite difference approximations 154; a( y. J' ^' ~" e
5.4 Consistency, order of accuracy and convergence 156
' \. A4 q$ b: n7 H# \5.5 Stability and the Lax Equivalence Theorem 157
9 ]  K8 W! n8 E7 ^5.6 Calculating stability conditions 160
6 g) h  _! |7 {4 h& |8 s5.7 Practical (strict or strong) stability 1665 N- V4 O3 s0 i# j
5.8 Modified equation analysis 169
4 v  O) s- a" O$ D5.9 Conservation laws and the energy method of analysis 177
+ B, ^; d4 `- H7 x0 ~5.10 Summary of the theory 186
5 ?- w! ^/ s: q! b3 M5 U9 BBibliographic notes 189' m6 S0 k+ J4 J( V8 ]" c
Exercises 190' f( C$ z& S) S6 H$ w
Contents vii
% j: g+ B  `2 b5 a, ~* y/ x6 Linear second order elliptic equations in
- V! o- n& h  G" y% K+ etwo dimensions 194
/ a3 ^. u) n) R, A0 }6.1 A model problem 194
6 R# }9 |; U# J: u8 g6.2 Error analysis of the model problem 1951 z0 _. s+ n* M& R5 P5 m0 N- d
6.3 The general diffusion equation 197
) P7 N2 F+ r5 P# ^9 p0 S( [6.4 Boundary conditions on a curved boundary 199
* D1 u6 |5 n/ s: p5 A0 d% ]6.5 Error analysis using a maximum principle 203
/ P' c0 ?4 U$ T. v4 s6.6 Asymptotic error estimates 213
9 S' x; z  H$ w8 s; G6.7 Variational formulation and the finite5 X, b: W# f8 E* k+ s0 z
element method 218; T" p6 ~; g4 y0 z* g; T: F
6.8 Convection–diffusion problems 224
+ Y9 X( g  W' K0 N7 i6.9 An example 228
9 Z6 P' q& O( ^. u+ N7 dBibliographic notes 231% K2 t; J1 e; N( J" ]7 [2 O! z6 V
Exercises 2323 L" P. L! y# W: T3 t
7 Iterative solution of linear algebraic equations 2357 T2 u9 k* O6 e+ o
7.1 Basic iterative schemes in explicit form 237
1 L2 ~' M* E! j2 u( J7 I5 N' E7.2 Matrix form of iteration methods and$ q4 V9 K* F8 Z" U) L
their convergence 2397 _8 `  ^+ m% A* C$ M5 u9 ]
7.3 Fourier analysis of convergence 244! d: u0 }# Q! w  U  m
7.4 Application to an example 248
/ O4 w) h/ t  ^7.5 Extensions and related iterative methods 250
+ M" S* l  o8 E: P, `7.6 The multigrid method 252) S+ W% W$ `$ w/ ?3 B
7.7 The conjugate gradient method 258
0 l8 c5 H( x8 L& g+ s; N+ \7.8 A numerical example: comparisons 261
, U8 _$ z1 E1 l3 IBibliographic notes 263* Q# ~  K7 M, _8 p, J2 \* `4 C# U
Exercises 2636 a- |7 [$ j) h( H7 t( G3 @
References 267
4 s& {% N; X8 O$ hIndex 273