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原版英文书第二版' Z  x6 r2 F  v- c
contents:
: Q' S% D: n* S( E  YPreface to the first edition page viii
4 U0 ]0 ]1 o6 h) ?Preface to the second edition xi
! \" I; E3 z; y8 K; r' f1 Introduction 1& [# H4 {& P# c( c
2 Parabolic equations in one space variable 7  G5 V" y8 E. S3 V2 f7 N$ q" S
2.1 Introduction 7
* j! o$ R5 U4 [( l( v5 v, c2.2 A model problem 7
1 |( j0 |/ Q7 m7 r( W- L9 v- S: a2.3 Series approximation 9
2 J/ t  h/ i% ~7 K1 p. K2.4 An explicit scheme for the model problem 10- k1 k9 P4 H' J4 a
2.5 Difference notation and truncation error 12* i# p- X& C, l' J
2.6 Convergence of the explicit scheme 16( v1 C- D% N5 T; Y
2.7 Fourier analysis of the error 19" r/ [0 E8 Z) P& U) Y% \$ J
2.8 An implicit method 22
6 V6 E9 q+ z. t# Y2.9 The Thomas algorithm 24! c/ c+ K7 j6 w  ~6 \' r/ ~6 ~
2.10 The weighted average or θ-method 26
9 |3 L7 W( T- t2.11 A maximum principle and convergence
5 G/ Y/ X6 k7 P7 e% T- Yfor μ(1−θ)≤ 11 B0 z3 O' B" g( e
2 33- A4 x# y$ Y/ |4 n6 U  @
2.12 A three-time-level scheme 38+ f( [* }/ ~+ i$ K  z8 x: B( ~
2.13 More general boundary conditions 39& B2 K) i9 [% Q* S& o
2.14 Heat conservation properties 44
: h% `* F& e2 U* i2.15 More general linear problems 46: j. w1 i* `) L2 p0 Y. i, ?  [
2.16 Polar co-ordinates 52
% H' ~% C2 ?) ]  Z2.17 Nonlinear problems 54
2 P1 T% ^7 G7 F% N4 cBibliographic notes 56; p8 O; M8 O3 J7 d
Exercises 562 P5 F3 H( _* C$ E( [
v  b* s* ], ^* Y; B
vi Contents
7 R* b3 c. l- s) \& @0 o/ G3 2-D and 3-D parabolic equations 62
; }9 k7 D+ N+ @3.1 The explicit method in a rectilinear box 62
, O7 M3 M; v' l4 I# L3.2 An ADI method in two dimensions 64: k6 U' O# j* V- r
3.3 ADI and LOD methods in three dimensions 70/ d1 e# A) j+ _
3.4 Curved boundaries 71
3 G$ `% l* x0 G  U; u7 w( R1 t3.5 Application to general parabolic problems 80
4 L/ r: u4 e0 b6 f& \5 V6 JBibliographic notes 83* M3 P' L$ b, {7 Q
Exercises 83
: V% Z8 N; o3 Y7 x8 g( W( A* A4 Hyperbolic equations in one space dimension 86
7 j8 f$ b! Y. |+ h! `4.1 Characteristics 86
" `, G# J: d- D2 ]. V% R; B1 P- b4.2 The CFL condition 89
/ M$ u9 h0 @- t$ b; U4.3 Error analysis of the upwind scheme 94- V- Q2 m3 r6 Y5 g* K, A
4.4 Fourier analysis of the upwind scheme 973 H2 @# ^" o# _. u$ Y0 s
4.5 The Lax–Wendroff scheme 1006 q* h2 S6 t# _7 _
4.6 The Lax–Wendroff method for conservation laws 103
2 O. v, f" ?5 y$ M0 `% w4.7 Finite volume schemes 1105 W7 P4 r. ]  u4 P2 u/ j
4.8 The box scheme 116
( q, f: c! i0 G- _1 a* F8 a6 |3 L4.9 The leap-frog scheme 123" u; e* f& W# v* M7 b8 s
4.10 Hamiltonian systems and symplectic
6 J6 n5 |, l) O6 J0 i1 M' iintegration schemes 128
+ y+ X1 `) @6 V: {1 N) l4.11 Comparison of phase and amplitude errors 1351 b4 `3 f6 l! i1 l
4.12 Boundary conditions and conservation properties 139
  r( O/ }  i7 X6 D8 J4.13 Extensions to more space dimensions 143
) e7 g+ M# }  G$ y4 k# @Bibliographic notes 146$ z" R" ]1 n/ ?" _
Exercises 1466 r- ~5 {; f* b  }
5 Consistency, convergence and stability 151
5 u  v$ Q9 m) H: l. C5.1 Definition of the problems considered 151
2 r, y, B( @  x' @0 E5.2 The finite difference mesh and norms 152
; x- e  J: H5 y5.3 Finite difference approximations 154
8 Z; O" B* m9 K2 F$ r5.4 Consistency, order of accuracy and convergence 156- V0 \- v; G  a2 s! n
5.5 Stability and the Lax Equivalence Theorem 157  ]- b( p, x6 {6 T8 L/ Y" [
5.6 Calculating stability conditions 160* p( P9 K! k8 ^- H
5.7 Practical (strict or strong) stability 166
( d; e- W% f) c7 R! m+ g5.8 Modified equation analysis 1695 \  U1 n% N1 y4 O$ A% J$ f% c2 {
5.9 Conservation laws and the energy method of analysis 177( S% z/ ^! t) E6 }$ p
5.10 Summary of the theory 1865 S- K0 t4 {6 g
Bibliographic notes 189( e, B5 p  u, F4 M7 C/ S
Exercises 190
/ j2 W) p6 f, o, u' ?* hContents vii8 i! J' G! H) B  H
6 Linear second order elliptic equations in
/ z8 [, ]) p% {4 mtwo dimensions 1948 ~7 [+ M( f* l% H  Y8 h# g
6.1 A model problem 194
! B9 b/ A0 P9 S1 K% Z/ r6.2 Error analysis of the model problem 1950 X! N" i8 n. r# U
6.3 The general diffusion equation 1979 ~  \& n' e9 O+ Z% ?: K
6.4 Boundary conditions on a curved boundary 199. N, x$ \" t. K1 W" o$ d: ^
6.5 Error analysis using a maximum principle 203; L4 o9 q! m( q& s* G. q* B9 Y
6.6 Asymptotic error estimates 213  y- z) T( Y' P
6.7 Variational formulation and the finite
5 ~0 F9 U6 G- m+ q, Belement method 218
; u* c$ R0 k; U5 m: x3 y4 `6.8 Convection–diffusion problems 224" o2 H5 e* ~9 P7 @' n
6.9 An example 228
! g% K8 T4 z6 N4 i) LBibliographic notes 231
: P: T# N+ i2 BExercises 232
" p4 k0 x) X* b" x% B5 R- d, d7 n* y7 Iterative solution of linear algebraic equations 2350 Z$ ?6 ?" _' u! L1 L) F; v/ O' a
7.1 Basic iterative schemes in explicit form 2375 t2 j6 N7 q  Z. d
7.2 Matrix form of iteration methods and, F# E  O6 E2 Z& \# T. _) W
their convergence 239) |: C8 g3 ]6 R) G% P5 L# Q7 ^$ J0 {6 N
7.3 Fourier analysis of convergence 244
/ G: `  N1 e- Q/ q8 f7.4 Application to an example 248
7 `: W) G+ G$ s1 m# G+ U7.5 Extensions and related iterative methods 250
6 e5 a; ]& o: J: k! d% t7.6 The multigrid method 252
0 O4 _$ D6 x3 n8 ~5 ?3 [& x# N7.7 The conjugate gradient method 2585 i  O& h% K8 s
7.8 A numerical example: comparisons 2613 e9 F$ D; {7 Y
Bibliographic notes 263- |' [. e  f" z9 D  y5 X! X1 R
Exercises 263. D- k8 ~+ j6 \1 Y& q9 B9 U0 H/ u
References 267
! }* G9 F& A* E0 H+ E. a0 {% S: _Index 273