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原版英文书 第二版
! G) s' T! B) N& S2 [contents:5 e% L# z+ n3 A
Preface to the first edition page viii; g$ t& w. T j# t! G
Preface to the second edition xi
5 F/ p6 v7 x [7 T6 W- w) q; J1 Introduction 1
/ L G2 A1 W5 {% k2 Parabolic equations in one space variable 79 _6 f/ _, f" l. y7 T4 z% d' z
2.1 Introduction 7
% j- e0 r$ e" N/ D2.2 A model problem 7
5 T6 T# |$ N7 }$ a" X2.3 Series approximation 9; J# z7 S/ f! K( `% g+ d1 q
2.4 An explicit scheme for the model problem 10
% ~$ U& K; Y" G1 ?8 r2.5 Difference notation and truncation error 12
7 H# P7 \; I9 C) ~ Y5 i2.6 Convergence of the explicit scheme 168 d2 ?9 \# Q& z: I
2.7 Fourier analysis of the error 19
/ b8 @5 g& b' j8 S6 N2.8 An implicit method 22+ N! W; W- L1 m) r- l
2.9 The Thomas algorithm 247 q$ t0 f' s4 @& c+ i( M! i6 @$ h4 z
2.10 The weighted average or θ-method 26
' U; \% Z) g/ ?5 k2.11 A maximum principle and convergence9 V4 h" ^0 v% `
for μ(1−θ)≤ 1
' d! _1 u( Y, ~6 P2 33
/ m3 n8 Z5 z* A& ~8 U( u7 G8 L2.12 A three-time-level scheme 38( {% M6 S( }" I$ q8 m7 H
2.13 More general boundary conditions 39
* ^: N5 ^, J6 E2.14 Heat conservation properties 44
9 E, {( j$ A9 x; E, E& y2.15 More general linear problems 46
: w S9 a" u5 Y7 E! F2.16 Polar co-ordinates 52
3 {4 a1 y& z7 L2.17 Nonlinear problems 54
+ E2 i8 N* ]' C/ U% n: |8 n% ]Bibliographic notes 56 w u: `+ @, C% @6 l! J' _
Exercises 56
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vi Contents$ G% g* D0 f% I( N
3 2-D and 3-D parabolic equations 62! n5 f7 M3 y- b' f: I2 ~
3.1 The explicit method in a rectilinear box 62
% K% Y* n6 j' {. g3.2 An ADI method in two dimensions 646 Z' h. j/ ]& c8 |4 U! Y
3.3 ADI and LOD methods in three dimensions 70
" W @% H" H S0 @6 u1 t4 X' j. I$ n3.4 Curved boundaries 710 C; F6 c7 n+ O$ O p# d
3.5 Application to general parabolic problems 80
- \% @: V+ G% wBibliographic notes 83' F. {8 f4 g; |2 C3 c& c
Exercises 83
2 o8 O# K0 E9 J- M* K& L4 Hyperbolic equations in one space dimension 86
" m% _! M# y% Z" \1 l1 z; C+ r4.1 Characteristics 866 [2 t4 R& A/ U1 j' ~- |6 o
4.2 The CFL condition 89
* r7 E* [0 {! c0 p# A1 {) j4.3 Error analysis of the upwind scheme 94
3 g1 g. r$ d2 ^5 Z' }7 c1 T4.4 Fourier analysis of the upwind scheme 97. q5 [$ t4 M' P7 s3 ~& p
4.5 The Lax–Wendroff scheme 100
/ a. i1 r0 C+ f: B. l4.6 The Lax–Wendroff method for conservation laws 103, c( x% K+ v! p
4.7 Finite volume schemes 110
* _: V) ~- o1 n9 e' y4.8 The box scheme 116: D4 d% ?. m+ u4 [5 b9 p5 T
4.9 The leap-frog scheme 1231 ?3 C9 {( D, r! n! z! U5 D5 l
4.10 Hamiltonian systems and symplectic S9 [8 a- }) V) x9 ^( \: H
integration schemes 128
5 E) [9 U+ X' I7 z8 V4.11 Comparison of phase and amplitude errors 135
2 k. x1 ^8 i1 K( i7 Z) E. r4.12 Boundary conditions and conservation properties 139, ]1 w9 `0 Z ]4 H3 t' @- \
4.13 Extensions to more space dimensions 1435 \$ H3 h+ [7 u8 d5 j
Bibliographic notes 146
4 p; C1 J% ~# _# VExercises 146
$ K+ F$ s& U1 N" I# j! B5 Consistency, convergence and stability 151- S9 G, k/ Y4 \, X, @. |
5.1 Definition of the problems considered 151
# b' ~# ^* i; X5 g. U9 x8 r% X2 q1 o5.2 The finite difference mesh and norms 152
, M: x9 A8 I7 u# T1 E# e. T' |( t5.3 Finite difference approximations 154
}% z& C- o: |1 W* t( \5 E. e5.4 Consistency, order of accuracy and convergence 156. d, q8 P8 y5 _9 e, l
5.5 Stability and the Lax Equivalence Theorem 157
4 \* g% r9 Y. _) l4 m7 [7 j5.6 Calculating stability conditions 160
% c& g1 a8 z& V9 D$ o5.7 Practical (strict or strong) stability 166
4 }; _% a4 I6 D$ t! p5.8 Modified equation analysis 169
" E: Z' n8 c0 p2 Y5.9 Conservation laws and the energy method of analysis 177
( o6 q' C) C3 @. n5.10 Summary of the theory 1867 @* ?" ^% c7 f. W0 H! s
Bibliographic notes 1895 U5 N. B; V' |- n4 F; p
Exercises 190, U! S& Z6 d- j" i" \7 J
Contents vii
! {' T# m3 ^# N% y5 p' W" f6 Linear second order elliptic equations in5 p) x$ ^; Z* J% K' H) a" B
two dimensions 194
7 V. I% c' i }) ]2 P6.1 A model problem 1946 k0 ?) j7 d* j) y$ ]6 W; \( ^
6.2 Error analysis of the model problem 195. x" g2 e3 D6 e
6.3 The general diffusion equation 197- B; r2 m1 y/ P& d
6.4 Boundary conditions on a curved boundary 199
6 C3 W) s' H; j- p' I* t6.5 Error analysis using a maximum principle 203
& N0 ]5 e% y. I+ X7 g5 W* t6.6 Asymptotic error estimates 213
, m2 V! q. i( C$ p6.7 Variational formulation and the finite9 R0 J0 b( l! T& B
element method 218! |0 @/ h0 e0 \7 d7 o" u7 t' q
6.8 Convection–diffusion problems 2240 C4 l1 M' j& }+ H1 ~
6.9 An example 228
7 u9 g/ O8 R: \Bibliographic notes 231% ?$ @+ Z2 z) z7 \
Exercises 232
5 F; N( V, L0 M$ Z4 a1 W F0 f7 Iterative solution of linear algebraic equations 235% m, A3 f& p" }: t
7.1 Basic iterative schemes in explicit form 237, x7 J6 v: p' s" O6 R) J
7.2 Matrix form of iteration methods and
; o* y, P1 f0 u7 e9 }their convergence 239$ ~$ }5 \; V b+ ?, D; N5 Q8 x
7.3 Fourier analysis of convergence 2447 R9 q* d2 V4 Z: {$ q. h
7.4 Application to an example 248( w( Y; ^$ L3 \2 D: g5 |
7.5 Extensions and related iterative methods 250, ]7 x* o Y/ o! W3 T0 N
7.6 The multigrid method 252
" B a" e5 |1 ~4 [7.7 The conjugate gradient method 2584 f3 {( ]7 s* ^ R, M$ U
7.8 A numerical example: comparisons 2611 V7 B8 `% g$ {2 m, e* h; ? W; c
Bibliographic notes 263
$ g3 A! W! {, Z+ RExercises 263
6 f0 X& q& J' D ]* tReferences 2673 V: K# k7 }' a9 f9 [; X+ x
Index 273 3 {2 Q; I/ C+ i% d6 z
- \0 A- i& D; m; a2 }7 z
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