原版英文书 第二版! m6 P, Q, G* Q; D2 C! y& E1 R) Q4 P
contents:1 p+ _5 N: b' {* ?1 x
Preface to the first edition page viii
, Z s1 I' W LPreface to the second edition xi
+ G4 w/ H2 a' V+ d1 Introduction 1
4 T. H; \5 K0 R+ D) w K2 ^2 Parabolic equations in one space variable 7
. m2 C9 F* P; {2.1 Introduction 7
s* T% h8 \/ f; H5 V- D1 E2.2 A model problem 7% t* o0 U0 A; c" H" V/ M% O4 R; q" O0 ]
2.3 Series approximation 9/ z- h! {! S# D
2.4 An explicit scheme for the model problem 10" V q& N, D/ P R1 S l& T) ]
2.5 Difference notation and truncation error 12; C4 o; \, W( d: J
2.6 Convergence of the explicit scheme 16 @/ B/ e/ I/ v! n
2.7 Fourier analysis of the error 19
; ^5 F9 T* J9 [8 ~2.8 An implicit method 22
) M- x/ V' \) [- `2.9 The Thomas algorithm 24
, F" q, Z2 ]1 ?2.10 The weighted average or θ-method 26
5 i2 Y8 M9 u+ t& I& _3 x2.11 A maximum principle and convergence
7 F: k9 j; A+ S+ Efor μ(1−θ)≤ 1% x8 j" s6 Y. c
2 33
3 f# Q: H3 e# @( B7 ]/ a2.12 A three-time-level scheme 38! g- n( C' I6 M: {
2.13 More general boundary conditions 39
! t1 R$ _* U3 c% d6 b. l# y6 r2.14 Heat conservation properties 449 k9 @1 j8 }0 L; L$ `: N) Y
2.15 More general linear problems 46
' h' U$ d7 p5 m& k S4 x2.16 Polar co-ordinates 52
$ `/ N, i! V( R6 M/ q% Y2.17 Nonlinear problems 54
2 e; s1 ~5 C& w+ }- R- tBibliographic notes 565 H* l3 Q R* d: \- u+ w: _: p/ l5 r
Exercises 568 [2 X$ X2 g. t' h" g6 H8 F
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! M; }$ @% Q. ^0 Fvi Contents
1 f. F3 M: U& E2 w3 2-D and 3-D parabolic equations 622 j3 d1 |! q, j4 C
3.1 The explicit method in a rectilinear box 627 H# Y% Z+ l+ J+ N; ^1 l {: f' U
3.2 An ADI method in two dimensions 64
2 j+ y" p4 Y$ E$ P% s. H( i3.3 ADI and LOD methods in three dimensions 70
2 w' n# j2 v9 Y0 E# E3.4 Curved boundaries 71
8 B7 y6 P- A( o* \. A# w3.5 Application to general parabolic problems 808 f# a$ B$ d! r3 S- T0 N
Bibliographic notes 83$ W- ^2 W9 j$ e# D7 }
Exercises 83" @$ J4 A9 m( X% i& X6 j
4 Hyperbolic equations in one space dimension 86* k* X- @' z/ A: M: j3 f
4.1 Characteristics 867 t1 C9 k9 B; J4 Y
4.2 The CFL condition 89
6 P- ~, c* g+ U% b' W& Q4.3 Error analysis of the upwind scheme 94+ O% l, @# F7 I; \9 T
4.4 Fourier analysis of the upwind scheme 97) H7 z1 G% d8 q( W
4.5 The Lax–Wendroff scheme 100. r5 E2 C6 n* k2 \! C
4.6 The Lax–Wendroff method for conservation laws 103( S# m* y! {% b$ [$ m; Q
4.7 Finite volume schemes 110
z) F# a. J, H3 y7 {4.8 The box scheme 116& x& B% B' h! F" E1 A- L
4.9 The leap-frog scheme 123
, e+ w6 t" J( p! \8 @& E( o4.10 Hamiltonian systems and symplectic
3 D s/ B" O4 Qintegration schemes 128
+ ?7 I7 r$ G# I% s; o9 b4.11 Comparison of phase and amplitude errors 135( B5 L$ f a3 x7 Z
4.12 Boundary conditions and conservation properties 139' x1 p5 p% H7 ~9 i. ?
4.13 Extensions to more space dimensions 1431 ^$ Y/ _. E* z2 N( w/ {6 G
Bibliographic notes 146
1 z! J2 _& ?8 w$ O5 J7 uExercises 146
+ D% W, s4 ^3 Y( h' Q g5 Consistency, convergence and stability 1519 b" [1 n0 p4 W3 L* u4 f
5.1 Definition of the problems considered 151" V9 j; M! |, F- H; ]; ~
5.2 The finite difference mesh and norms 152
- a7 i/ x1 S: F$ ]5.3 Finite difference approximations 1541 d; G; X# f1 S4 m( L
5.4 Consistency, order of accuracy and convergence 156) s r* n0 i% P; o+ S3 E- z) R
5.5 Stability and the Lax Equivalence Theorem 157$ m8 x8 a6 C5 ]3 s) H
5.6 Calculating stability conditions 160
9 P$ B# S* G0 u, ]. j1 s/ U5.7 Practical (strict or strong) stability 166) {# q9 o7 _* o- _5 \
5.8 Modified equation analysis 169, t0 Z3 X/ _% X/ }2 \. u
5.9 Conservation laws and the energy method of analysis 177
/ ]; s) ~6 A- k5 @5.10 Summary of the theory 1862 W, J. {5 W; W
Bibliographic notes 189
- V8 {3 p6 P; L% E/ EExercises 190
P) H! H6 x1 [' b/ ]Contents vii
7 m E' Q7 e; f* t0 s' Z6 Linear second order elliptic equations in
9 |! t5 l. c! dtwo dimensions 194
" P" I, n4 [2 i- n0 B) X0 Y- |6.1 A model problem 194
1 S2 p5 [1 s1 r& f6.2 Error analysis of the model problem 195
6 e- ~+ ^& O, S" r0 m9 B6.3 The general diffusion equation 197
Y* \& ~4 y- k$ W# I3 ]6.4 Boundary conditions on a curved boundary 199
6 J+ B5 x$ l' R5 V) u# H# Z/ d1 M. i6.5 Error analysis using a maximum principle 203
" ]" X7 c6 G, Y5 X" o W4 o7 g6.6 Asymptotic error estimates 213
+ {9 T* R$ u) N% I3 `' d$ u6.7 Variational formulation and the finite' E% d+ _; W2 s: c: o$ }1 d: H
element method 218& y8 B, x3 C0 C) h) D0 O3 k
6.8 Convection–diffusion problems 224
* j4 j6 }' }& }. i6.9 An example 228
, O- D J" z/ CBibliographic notes 231% q" I* m) j0 r- _5 W; i. v; t
Exercises 232
8 w; ]2 }6 j, j4 g' p+ y7 Iterative solution of linear algebraic equations 235
* r9 `# b C5 i; z( K7.1 Basic iterative schemes in explicit form 237
6 G3 O5 e: l6 D5 b4 |) t7.2 Matrix form of iteration methods and
( L2 R1 P* M8 f$ W) |their convergence 239
, [9 t5 ~. i e. A3 _1 L7.3 Fourier analysis of convergence 244# ^$ y& ^$ w4 ~. E2 J
7.4 Application to an example 2484 F9 F' f8 t X U# @
7.5 Extensions and related iterative methods 2505 k9 }# p+ f/ \# Y. f' o& z
7.6 The multigrid method 252
Z$ Z, l% R: T7.7 The conjugate gradient method 258- E2 P7 H- f( W) ?* ?' ?' a
7.8 A numerical example: comparisons 2611 H; I, y$ r" q/ {
Bibliographic notes 263: x. e( `! ^( |9 [' P, k
Exercises 263
! D! h8 Q' b& ~5 I5 k4 P. y9 fReferences 267
$ H/ H! N- M QIndex 273
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