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
+ j: B3 V! N4 T) vcontents:
& S2 e/ F, ^: o4 `# e( Q9 T7 HPreface to the first edition page viii
3 A5 U$ N4 q- c9 ?: z# pPreface to the second edition xi  v  @* d. l: T2 J
1 Introduction 1. l, ~7 o  i8 {- C% ~
2 Parabolic equations in one space variable 7
0 I+ b9 B, j" v" f& F: R2.1 Introduction 7! w  }; Q! K  L" }# q2 o) @* u
2.2 A model problem 7
$ B, t7 T* n& t2.3 Series approximation 9" g! w9 _6 n8 L6 s/ |/ H2 \  l- p
2.4 An explicit scheme for the model problem 10
& ]" A/ R1 E5 }0 n; `2.5 Difference notation and truncation error 12; F1 T+ g/ J, O3 r' A
2.6 Convergence of the explicit scheme 16
3 f8 `8 W. C: ^. J6 F6 V2.7 Fourier analysis of the error 19
+ w7 Y; L$ P0 y( W; E) Y1 N; a2.8 An implicit method 22
% v+ T. x6 O7 t' T2 b7 h2.9 The Thomas algorithm 24( ?5 R9 W1 K6 M4 w: T
2.10 The weighted average or θ-method 26
4 ^" ?, M7 H/ R% @2 z9 a0 k2.11 A maximum principle and convergence- _! q6 }* }: I0 ~. h
for μ(1−θ)≤ 1
; M* z1 Y. ]7 A, H( `  B. ^2 33+ E6 C. t) Z/ Q3 R  l3 J
2.12 A three-time-level scheme 38
* w; D8 Y, p7 M8 E3 ~2.13 More general boundary conditions 39
+ L4 a! S- w# V( r! b! h7 }2.14 Heat conservation properties 44
# I5 R; O' U3 E  {" o2.15 More general linear problems 46
: t: Q3 B. n( @* o3 f2.16 Polar co-ordinates 52
1 v. b/ i7 K/ ^0 F$ Q2.17 Nonlinear problems 54, g" F# @  _6 q8 I9 u/ D! X
Bibliographic notes 561 l7 f4 g: v0 Y7 G- O
Exercises 56
8 K1 v5 g+ W* u$ v  f* Iv
* G$ Y; V. Y2 ?vi Contents: X8 `# x4 U+ l3 X, n
3 2-D and 3-D parabolic equations 62& t, |5 j  L2 P/ t9 J
3.1 The explicit method in a rectilinear box 62
2 m1 U) L# ^+ I% N3.2 An ADI method in two dimensions 64
; F1 S. Q9 g! D, F7 ~3.3 ADI and LOD methods in three dimensions 70
% @* r6 A# c( W3.4 Curved boundaries 71
8 H7 [# T6 X7 B* @) \5 e3.5 Application to general parabolic problems 80  D' F, Z& y, w  K
Bibliographic notes 83
2 T# L- m8 p* W* w: `Exercises 832 ^8 I/ U4 i3 }
4 Hyperbolic equations in one space dimension 86
! o% K2 [3 M# t( y! R4.1 Characteristics 86
/ Z: F9 H3 }4 b: z4.2 The CFL condition 89
( O, M  i# H) s% X4.3 Error analysis of the upwind scheme 94
' Y7 |$ N$ D' h1 f& \$ i; K; Y4.4 Fourier analysis of the upwind scheme 97
) ^/ s% _5 }6 ~4.5 The Lax–Wendroff scheme 100. r3 r" Z7 K4 k6 V6 \  |( v4 ^, t
4.6 The Lax–Wendroff method for conservation laws 103# J- q, ~# M+ W( \* u* U9 Q
4.7 Finite volume schemes 110
) v' g+ }( e2 d. N4.8 The box scheme 116
& M( u) V! `4 |- ~% I& J4.9 The leap-frog scheme 123$ O5 Q5 C  p/ E8 h5 ?
4.10 Hamiltonian systems and symplectic
6 I0 H: I8 S# @" [) B5 \: u! V" yintegration schemes 128$ t3 O. _& d5 u: s+ n4 m
4.11 Comparison of phase and amplitude errors 135
: I2 B: Q' }) D2 S4.12 Boundary conditions and conservation properties 139: j; b% r& b5 h) h& X4 ~4 F
4.13 Extensions to more space dimensions 1439 E0 \$ x' q$ B. g
Bibliographic notes 146) a( s+ ^1 j; U; N: \. ^" ~
Exercises 146
( G$ ^& X; u3 K4 P5 Consistency, convergence and stability 151! x- O: W" ~# @# C9 t
5.1 Definition of the problems considered 151
2 c4 [7 u+ @: v1 _4 Z: T* Z  Z5.2 The finite difference mesh and norms 152/ ~: H8 W1 \6 B& `  ^: r1 y- a
5.3 Finite difference approximations 154
8 B# d5 H2 E. \; H7 m& f( u5.4 Consistency, order of accuracy and convergence 156
, V: @0 _3 `8 R- R/ D' E: o5.5 Stability and the Lax Equivalence Theorem 157. ^7 D1 N. _1 ]% ~6 l1 K, i
5.6 Calculating stability conditions 160
5 I8 u0 ^; ]3 i: R+ x1 {5.7 Practical (strict or strong) stability 166' `. u6 j6 X% {; s, T: l
5.8 Modified equation analysis 169, F0 ?9 }- t: w2 \: C" e
5.9 Conservation laws and the energy method of analysis 1774 F+ M" [. s! `/ C! ~
5.10 Summary of the theory 186
2 @7 x" O+ N  a; @0 c1 i2 L: c2 nBibliographic notes 189
- I/ Y+ D! ?' ~3 \Exercises 190
% Q7 E( b" h- Z) JContents vii
4 M! Y* O/ X5 g4 y6 Linear second order elliptic equations in5 r3 N9 x2 X: q3 G3 Y( k" h
two dimensions 194; M3 b" M7 N/ v  ?3 \% Q) l( o
6.1 A model problem 194
" l2 j6 h+ g9 Y8 J% j: y6.2 Error analysis of the model problem 195
8 ?/ I# [6 e& n6.3 The general diffusion equation 1971 E+ d8 w1 {4 Y5 |- U  P" ~! [$ G8 F9 c
6.4 Boundary conditions on a curved boundary 1990 k/ _4 q6 x+ W4 \) c
6.5 Error analysis using a maximum principle 203
3 o* q! d8 p8 b! ~& {* @6.6 Asymptotic error estimates 213
4 H% k, I2 t# V& c6 V7 A% h( n+ B1 c6.7 Variational formulation and the finite
# c9 u. }* \- Xelement method 218, I3 }1 A2 c: R
6.8 Convection–diffusion problems 224
- }5 F, P2 X, L  s) f- K6.9 An example 228  T  h9 a6 P! }, G# `7 d
Bibliographic notes 231
& ^& J  x+ U4 I% ?; {% O: Q/ `1 N' ZExercises 2323 j: |; ?9 B2 c4 k* y! q0 u
7 Iterative solution of linear algebraic equations 235
  A7 l8 [5 k( U- Q7.1 Basic iterative schemes in explicit form 237
; m9 B' w* y. g" g& @# i* k, f7.2 Matrix form of iteration methods and
" M! w0 [. a( htheir convergence 239) ?7 Y9 O+ i( H5 q' k2 d  i
7.3 Fourier analysis of convergence 2441 W8 f2 K, Z# d! @) S" H) g, i
7.4 Application to an example 248% F) E4 }. x# `, J
7.5 Extensions and related iterative methods 250
! {) w; p5 r2 }9 C% `, D. \7.6 The multigrid method 252
/ U+ {/ H, D4 E. c$ `0 p% b+ A7 D7.7 The conjugate gradient method 258
* b2 _$ Y( v/ F' ?# o; D# ^7.8 A numerical example: comparisons 2611 y8 e5 ?7 M5 K2 ^$ I2 r
Bibliographic notes 263
) U# M* E/ C; t8 BExercises 263! J+ f: ^; W" j1 o5 H9 u+ Y
References 267* N: {) v. `5 X! U, n7 Z
Index 273