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
( a$ h6 C4 C4 z9 mcontents:1 A  x% Y4 V: r! x/ E4 C$ S) w/ @3 n% |
Preface to the first edition page viii1 K' U0 W$ O$ d9 w+ A1 W
Preface to the second edition xi; e6 J5 p9 N2 B2 o2 S3 I& }( g
1 Introduction 1
; d8 z2 L2 `2 V: P- I$ R2 Parabolic equations in one space variable 7
+ ~% Q. F$ F' a  L0 w" K. U2.1 Introduction 7* |7 t$ J; d4 L1 C( j/ e9 v
2.2 A model problem 7
! Y0 v; `. ?* R: q8 @. ]2.3 Series approximation 9: |8 a0 o5 ^) A
2.4 An explicit scheme for the model problem 10( C# K( C/ w/ j, g
2.5 Difference notation and truncation error 12
8 I9 X& X$ g7 S( q$ C2.6 Convergence of the explicit scheme 16
8 j" `1 f+ x, D7 r. k) ^% R2.7 Fourier analysis of the error 19
6 I7 J+ g$ W9 A3 d* I- Y$ m2.8 An implicit method 223 I& F  |7 h& f/ o% ?7 d3 q- [( F, \
2.9 The Thomas algorithm 24
7 E. o5 a4 }1 l; P& `3 N. i9 d2.10 The weighted average or θ-method 26
% X0 Z3 N4 A  P4 X' s1 _2.11 A maximum principle and convergence1 P/ g  P  h( q: I" C) [
for μ(1−θ)≤ 1
' N3 ]- ?1 @  r2 331 N) P* {9 k% k( _+ L$ P6 b# e
2.12 A three-time-level scheme 38
- `" c" F+ U& }2 I1 j) J( B& Z2.13 More general boundary conditions 399 h, P7 N/ C- @$ m. J
2.14 Heat conservation properties 44
  K. Y: A# k) q7 m6 [% i, A2.15 More general linear problems 46
' X4 f& p9 ^0 ^! W# X' ]2.16 Polar co-ordinates 52  Z& R8 w. E1 {. }' ~7 W% _* B
2.17 Nonlinear problems 54
3 K7 N' r& l+ F. c9 }Bibliographic notes 56* f% \9 E; n- J7 W
Exercises 56
6 c* |2 q4 K6 d. Jv; d  ]3 U1 \+ l0 z4 Z& r
vi Contents
* @: n5 v# A+ [3 2-D and 3-D parabolic equations 62) W6 D9 w' @7 ~7 J( w$ ~* H& r3 J/ c  H
3.1 The explicit method in a rectilinear box 62  ]2 X6 x( v, m! H" A7 j
3.2 An ADI method in two dimensions 646 Y8 x0 D" x) n" }( @
3.3 ADI and LOD methods in three dimensions 70
* T, l( s6 W4 F; |' d7 ~3.4 Curved boundaries 71# _+ g% d) G8 J! Y6 `7 o+ `
3.5 Application to general parabolic problems 80
- N5 T$ O% Z  s0 i3 F1 nBibliographic notes 83
. S2 w& W4 `1 D' q: [- I3 B( eExercises 83
7 a2 v+ G; c/ N- Z& N3 A5 ?8 L4 Hyperbolic equations in one space dimension 86
( U! M( n/ l, V0 D4.1 Characteristics 86
  l$ V7 b  N) E; g9 Z* Y& X# w1 t4.2 The CFL condition 89
/ K! ]1 @5 ^. v0 w: O1 a0 T4.3 Error analysis of the upwind scheme 94+ F8 n. B0 [6 b  g( N" Q' s3 U4 x
4.4 Fourier analysis of the upwind scheme 97' g9 S; j* V& s1 J. @8 e: p4 P
4.5 The Lax–Wendroff scheme 100+ e' Y; n% V# X* E( z7 h
4.6 The Lax–Wendroff method for conservation laws 103  E! ]' Y# ?- J
4.7 Finite volume schemes 1105 z" M" b; I6 ^& U5 L0 B* u# Z( C) V" ]
4.8 The box scheme 116, O- ~7 j0 H- D1 H
4.9 The leap-frog scheme 123  S  [" E' ?8 p4 P& i+ I
4.10 Hamiltonian systems and symplectic
6 [& }2 R: O. _) @2 `, K9 r- F* z0 Sintegration schemes 128
: q3 t* ?8 k/ c1 s( G4.11 Comparison of phase and amplitude errors 1357 L, O$ A' H8 C1 R9 \6 W1 g
4.12 Boundary conditions and conservation properties 139
3 n& f/ B0 \; ]0 p4.13 Extensions to more space dimensions 1437 x4 P) }# F1 _/ F) Q1 z" I- P
Bibliographic notes 146
5 H& {1 g6 b4 _. Q& k* n( _( y" GExercises 146
  G+ p* P  ?9 G: J+ }  f5 Consistency, convergence and stability 151, s$ M( `. Y+ U/ q* i9 I
5.1 Definition of the problems considered 151, C1 w4 `3 t0 l& j! I: b0 z
5.2 The finite difference mesh and norms 152
! u! V9 K2 p# l" R) c6 U/ Y" Q5.3 Finite difference approximations 154
- [  C7 F$ J# n5.4 Consistency, order of accuracy and convergence 156
; t, t" t7 M; u) Y. \+ z" y3 J1 I5.5 Stability and the Lax Equivalence Theorem 157
. {: _( ]% ~) X5 g( y2 x" b5.6 Calculating stability conditions 160
0 L2 k% L! u+ k7 I5 N4 W8 O1 `5.7 Practical (strict or strong) stability 166
: B2 A4 V4 q; t9 Q; P$ H+ w' F! y5.8 Modified equation analysis 169
3 x$ O2 F; r' I, ~: x) n5.9 Conservation laws and the energy method of analysis 177
* A0 ?- _& j1 y" v5.10 Summary of the theory 186
+ Q2 u  o0 m# U( v* uBibliographic notes 189
2 J! i' K& J* T1 W( GExercises 190
7 W, x2 a9 D7 v6 R- N/ a' bContents vii7 I- Z+ l9 ?0 r
6 Linear second order elliptic equations in
" b" F! Y* o& g- B  `two dimensions 194
0 t) K- K" A, K9 @$ e9 Y6.1 A model problem 1940 U) t+ G) b3 c; s) _3 D# p# j- ~6 H
6.2 Error analysis of the model problem 195+ o% Y- V+ @3 _/ ?
6.3 The general diffusion equation 197& m+ h2 {+ I: D4 }+ C) R, f* r2 P
6.4 Boundary conditions on a curved boundary 199
2 s2 {2 d8 E+ D4 L/ q* \* f! T6.5 Error analysis using a maximum principle 203. S; g$ b9 }3 ^) O) t9 n
6.6 Asymptotic error estimates 213* h5 L- L3 e( E9 y* k6 Z/ t
6.7 Variational formulation and the finite  @1 }3 r9 ?# M& q" ~0 m
element method 2188 W3 A! [3 r  Y' {
6.8 Convection–diffusion problems 224
$ c8 g5 k, @# G0 H5 E6.9 An example 228) S6 K+ f0 Q; k
Bibliographic notes 231
0 x0 P$ |) Q- F$ AExercises 232
0 U  o9 Y. n7 t4 G) o7 Iterative solution of linear algebraic equations 235: Y4 I; b2 W$ B( Q5 T4 g
7.1 Basic iterative schemes in explicit form 237* W5 y/ z5 i6 Q* @& ^
7.2 Matrix form of iteration methods and
9 b$ J( v1 O) ]- Btheir convergence 239' _1 K; p) z% [8 A+ n5 \% M
7.3 Fourier analysis of convergence 244
9 \6 F. e7 c) q4 z' q. @; ?" L- P7.4 Application to an example 248
, R; b& c2 q8 k& W2 Y" f8 V. l: |9 v7.5 Extensions and related iterative methods 250
% J% ]) `" D2 e7.6 The multigrid method 2520 T5 _# T3 U/ Q7 r" [; k. Q
7.7 The conjugate gradient method 258
. y1 |; Z& R  u* j9 `1 W5 E4 X7.8 A numerical example: comparisons 261
& M# L( ~+ Y- j0 W* A9 Y# qBibliographic notes 263. \* [% j, Z0 Y' H& C% e2 Z) K
Exercises 263
" Q( }0 n1 u2 ^: b3 \References 267
4 w( {: ?- U0 M; V& QIndex 273