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原版英文书第二版0 D- P1 l0 O. Q( @$ R" J$ \9 b
contents:
! h5 X9 M+ ^$ c( [! ?1 b! b! DPreface to the first edition page viii
( @$ b; i: b) Q# _6 w& UPreface to the second edition xi' U( M; y0 K/ Q
1 Introduction 1
0 L$ F1 ?+ L  x% j9 B2 V% ^( {" V! p2 Parabolic equations in one space variable 79 q# K. z3 S4 {& D2 A
2.1 Introduction 76 N+ U" |8 Q/ w, Z8 R
2.2 A model problem 7# U! w* H) Y& T% t
2.3 Series approximation 9
; b5 Y8 X9 H' {$ q: K2.4 An explicit scheme for the model problem 10
: v" H( [1 g1 X. A. E2.5 Difference notation and truncation error 12( q5 s1 b' b- v6 a4 S& r
2.6 Convergence of the explicit scheme 168 C' }0 m6 K" A3 |# i2 s( @( `7 i
2.7 Fourier analysis of the error 19, p" N8 W$ s$ v& g1 K- v
2.8 An implicit method 22
- h) R8 |5 e8 S2.9 The Thomas algorithm 246 W# E( R* J2 {4 a* i, b$ n" E
2.10 The weighted average or θ-method 26. ~# u% q. L, R2 A6 ?
2.11 A maximum principle and convergence$ K5 `- o4 ]9 s
for μ(1−θ)≤ 1" @# X' e( L. i
2 33( S* h4 X$ S" ^$ a" B
2.12 A three-time-level scheme 38
" s0 f" Y( k' J2 W- K' S2.13 More general boundary conditions 39. H/ j; ?/ F. N
2.14 Heat conservation properties 44
9 ~7 a* [# D% @! D/ A2.15 More general linear problems 46
9 t, e' y; `( a4 U2.16 Polar co-ordinates 52! g$ A3 C2 J4 j! [6 F
2.17 Nonlinear problems 54
5 b: W+ _8 e8 FBibliographic notes 568 b, E+ d7 \- D  ?6 ?
Exercises 56
/ X- ~, h; M7 O0 b/ sv* z% C5 [/ {% O* J
vi Contents- a- S% \: t5 s* @2 g, E
3 2-D and 3-D parabolic equations 62$ M9 d7 K6 n7 h% N' t
3.1 The explicit method in a rectilinear box 627 ^  E* l' G: o2 Q1 H. H, u% H
3.2 An ADI method in two dimensions 64
$ A% b, k: `( ?3.3 ADI and LOD methods in three dimensions 70- @- ]' S- ^% \
3.4 Curved boundaries 71
' O' r+ P% C1 F4 N* q3.5 Application to general parabolic problems 80
/ |& K; D. g: n7 z8 ~Bibliographic notes 83
% W4 W8 G- F9 v/ j9 Z# S* kExercises 83
6 t2 I4 l5 S$ o7 `4 Hyperbolic equations in one space dimension 86! x5 q. j0 ?' D& B/ t
4.1 Characteristics 86( m0 C/ m: ?2 z2 R
4.2 The CFL condition 89
$ x1 m: \4 @" O9 |: ^) D: P4.3 Error analysis of the upwind scheme 94
/ m2 s( s# ?) p7 I# Y! u% Y4.4 Fourier analysis of the upwind scheme 97& ]: R! Y* a4 A5 b
4.5 The Lax–Wendroff scheme 100
/ x# k1 c0 N9 J+ G- Z: {4.6 The Lax–Wendroff method for conservation laws 103
+ t. o( e  p. v7 b3 P/ P9 m4.7 Finite volume schemes 110
' k( D4 @( ~# C! i! p3 o4.8 The box scheme 116, T5 ~4 K% M2 p. F
4.9 The leap-frog scheme 123! Y- n, n4 M0 d$ G
4.10 Hamiltonian systems and symplectic: K# U( j' O) K; H, S) E
integration schemes 128
+ O7 t! ]8 l0 j% d+ X# Q6 q4.11 Comparison of phase and amplitude errors 135
2 t3 h9 P, i( [% w4.12 Boundary conditions and conservation properties 1393 S4 B! Z. T, E) O6 t0 R
4.13 Extensions to more space dimensions 143
8 O2 h. v4 R6 U4 ^0 s. _: N4 y5 FBibliographic notes 146% n2 f% a2 B  v' v) j; j
Exercises 146) Z" P  _* \# o- W9 s1 J
5 Consistency, convergence and stability 1515 z0 U# P5 @6 n1 P" M5 V
5.1 Definition of the problems considered 1515 Z2 o' g* o- D5 g
5.2 The finite difference mesh and norms 152, K/ \$ q% z6 _
5.3 Finite difference approximations 1542 {) I$ A  g5 N3 P1 _
5.4 Consistency, order of accuracy and convergence 156
& V- L$ o( R/ T5 q# a5.5 Stability and the Lax Equivalence Theorem 157
+ X- }3 t0 K8 q' @( g5.6 Calculating stability conditions 160& ~4 F' T" O% b$ F1 m( T7 W) Z" F. O
5.7 Practical (strict or strong) stability 166
$ S; g' q9 {: f# J" p1 a5.8 Modified equation analysis 169% ^/ h' S* k& F+ |' z7 Z7 k; u* S
5.9 Conservation laws and the energy method of analysis 177" I; `# Q& ]6 Q% I- C
5.10 Summary of the theory 186
6 G' @- j2 Q: o( R! [8 \- WBibliographic notes 1891 V1 G0 l* [0 `* ^5 x; t
Exercises 190+ ]; `5 a  E% \
Contents vii+ h( Q, X! S- p6 u# e2 ?. Y! N8 P& i
6 Linear second order elliptic equations in7 H3 D! V. z! L0 e! {
two dimensions 194  l" P7 z, u  m6 l6 o& O$ L( m
6.1 A model problem 1948 x1 X& d/ h( w5 [. c$ k, z7 X( c: d9 l0 ]
6.2 Error analysis of the model problem 195
  j! \6 j2 M# t+ v) N8 q( y6.3 The general diffusion equation 197! Q  b0 u! q3 V4 x* W
6.4 Boundary conditions on a curved boundary 199. z) `3 Z7 l' Y7 C1 n% d. F
6.5 Error analysis using a maximum principle 203& H+ `7 O! @6 ~0 Y
6.6 Asymptotic error estimates 213
1 P7 E. q) X+ H6.7 Variational formulation and the finite; A* X+ d+ R  q$ i- ^  c
element method 218
* j) z( {  T# k2 L3 z6.8 Convection–diffusion problems 224
. T! E7 F& G9 l3 t6.9 An example 228  @$ P# b8 _: g  f# X
Bibliographic notes 2317 F3 g7 r, X, Q" N
Exercises 232
1 m5 J/ D2 n8 F/ g' O' B7 Iterative solution of linear algebraic equations 235
" A# s* a" i- X6 t! L' ~7.1 Basic iterative schemes in explicit form 237: l: G1 R6 y* k# m4 t
7.2 Matrix form of iteration methods and
9 E5 q1 C; y& [! R8 h1 V' i4 Mtheir convergence 239
9 i! L" ^$ u* C3 V7.3 Fourier analysis of convergence 244
6 t/ e# C# i  q7 j2 R; y7.4 Application to an example 248
7 v2 u+ Z+ d8 k! z# G+ V, x7.5 Extensions and related iterative methods 250
0 Y; [3 U$ N/ {' w- P! o7.6 The multigrid method 252% f- h1 Z5 n6 R- M0 N
7.7 The conjugate gradient method 2586 J, Y- J1 ]' f, J
7.8 A numerical example: comparisons 2616 h# T1 o+ B/ c4 L1 R' ~
Bibliographic notes 263
" d& [: e; j. p) G9 ^1 ZExercises 263
6 m8 a  `4 n: L( V% ZReferences 267& i. `( o! {. ?7 \( u
Index 273