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原版英文书第二版9 ?% j* ^4 q& F: w" I
contents:
$ K/ b* j4 @( G' y# t1 KPreface to the first edition page viii' {9 _# C6 T5 z- l5 o: h5 t4 k
Preface to the second edition xi
- q, {& e8 M% b6 O0 t1 Introduction 1
' S+ K! X: O6 @/ R2 Parabolic equations in one space variable 7" M! x9 U$ X2 V. n% y
2.1 Introduction 7  H7 C2 g) `4 ~  J
2.2 A model problem 7
' f8 g" K$ A1 x, E" e9 ]2.3 Series approximation 9
( U# y$ p* v( F1 C4 ?9 {1 i2.4 An explicit scheme for the model problem 10* n6 D, J8 Y2 L/ {
2.5 Difference notation and truncation error 12- ]/ |+ F: n6 d& w! O8 W9 q
2.6 Convergence of the explicit scheme 16
& s) v% B- f- V+ `  a; m2.7 Fourier analysis of the error 19# L, Q2 j5 A3 f, r; n' x
2.8 An implicit method 224 S7 d$ O6 M& z/ B
2.9 The Thomas algorithm 24
4 @3 Q. J( |0 u8 e) D" v" G6 v2.10 The weighted average or θ-method 26! ~; B: h2 o; [7 ~
2.11 A maximum principle and convergence  }0 {9 h) ~, `! i' {' r
for μ(1−θ)≤ 1
2 s$ Y$ r. l3 `/ F2 33; F( s" y: M! D- w3 p
2.12 A three-time-level scheme 38- M7 j3 p; u- n  b! k' V6 s3 `$ G
2.13 More general boundary conditions 39
/ }0 _# D/ F- m; A/ a2 Z2.14 Heat conservation properties 44
8 t6 d- x, f3 ^4 p& Y* [0 Q! W2.15 More general linear problems 46
5 j) k. N) V$ A2.16 Polar co-ordinates 528 K- F! ~5 l; c' D. N1 G
2.17 Nonlinear problems 54
; _& _& Z; T) J$ N( kBibliographic notes 56
9 s/ X' U- b' g- x0 pExercises 569 S. O2 t6 a: P8 s
v2 d, F+ F& B- q4 `4 R& n0 g- _
vi Contents6 V7 O$ F" z% w% v! ~' C  w) z1 D* ~
3 2-D and 3-D parabolic equations 62
& u1 |9 a7 L" |+ R5 v3.1 The explicit method in a rectilinear box 62
. N# Q7 d, t7 u( C* B3.2 An ADI method in two dimensions 64) W7 e5 L8 \0 c* J% \  F' G
3.3 ADI and LOD methods in three dimensions 70
! @, ?  a$ g5 j0 j- O) K6 T3.4 Curved boundaries 71
4 J/ A6 g. G! r8 B3.5 Application to general parabolic problems 808 |; h4 L7 K+ Y( P0 n- H* Y
Bibliographic notes 83: Y0 N( F* K9 n0 b
Exercises 83) b" v& o3 C; }+ m
4 Hyperbolic equations in one space dimension 863 o8 k( q* R5 p* @
4.1 Characteristics 86; @6 `/ b. S# E
4.2 The CFL condition 893 W7 n5 S' s' B9 z
4.3 Error analysis of the upwind scheme 94
) x. L( {8 f% R4 `0 w4.4 Fourier analysis of the upwind scheme 977 l, t5 T4 }8 G& S/ x/ L4 R2 N0 h
4.5 The Lax–Wendroff scheme 100( f) L4 Z& O* T
4.6 The Lax–Wendroff method for conservation laws 103# o3 K1 E0 S! g- k, W
4.7 Finite volume schemes 110
6 N  _& m  g, ?7 [" x$ C4.8 The box scheme 116( u+ c1 h$ w0 \  ~& Y+ D4 \  u
4.9 The leap-frog scheme 123) h( |, o1 i) M* ~" g  E- X
4.10 Hamiltonian systems and symplectic" L" r% @! q" Q
integration schemes 128* I, i$ t  q3 w7 h
4.11 Comparison of phase and amplitude errors 135
! E: H$ q, r* N0 e4.12 Boundary conditions and conservation properties 139
2 K, }+ M$ d' g0 g4.13 Extensions to more space dimensions 143
6 B8 N0 i* `6 ?$ tBibliographic notes 1469 p. o3 I5 j* Z: }% B4 T$ F
Exercises 146
& Z1 l" m4 Z: y. b5 Consistency, convergence and stability 151
8 f5 m7 a3 l2 u6 V7 ~1 N& Q7 [9 W& ^$ U5.1 Definition of the problems considered 151
5 \. P6 q: x8 m4 {4 k& N5.2 The finite difference mesh and norms 152" ]! p4 Z: R% B6 D- f! ~& @
5.3 Finite difference approximations 154, ^% U$ y' ]$ p( Q0 s
5.4 Consistency, order of accuracy and convergence 156+ P4 F+ @& I5 U3 I5 F
5.5 Stability and the Lax Equivalence Theorem 157- Y: M" ]% v" d7 ?1 C! u
5.6 Calculating stability conditions 160
0 m. e+ s) L* ^4 U( p* O6 [5.7 Practical (strict or strong) stability 166
. ]4 c! U9 m; u) @4 u5.8 Modified equation analysis 169# w2 R, Q- G* K: j, i, x
5.9 Conservation laws and the energy method of analysis 1778 {! ~5 Y! K( t- q' f* b% K6 p2 @
5.10 Summary of the theory 1863 ?- T4 i* l/ |! u/ V- Z; s' e
Bibliographic notes 189" q& n! n5 y: S* E: \
Exercises 190
9 e. p5 K/ m$ V: }Contents vii
8 M4 f5 U* v0 H6 Linear second order elliptic equations in
( u# L0 p3 j# n" D, H( Ktwo dimensions 194) [4 V3 r: c& ^) D
6.1 A model problem 1946 c' J& h: |2 O
6.2 Error analysis of the model problem 195
5 M1 X$ m0 V8 z4 _% E8 N/ C- a6.3 The general diffusion equation 197
8 U% C& M8 T, u" Q6 K1 y6.4 Boundary conditions on a curved boundary 199
7 l9 a4 J6 T& M7 n& b5 h6.5 Error analysis using a maximum principle 203, {' p4 N8 K+ b, Z; M; c1 J
6.6 Asymptotic error estimates 2139 W1 \. W, ^; Y4 y1 n0 x
6.7 Variational formulation and the finite# e6 T/ C8 R+ C5 m
element method 218. |1 r3 Z* L  I
6.8 Convection–diffusion problems 224$ o" X' ~" o6 c1 @1 W0 d9 Z2 E* E
6.9 An example 2282 g/ b' e* x& V' r2 g3 o1 q3 ?; a
Bibliographic notes 231+ z' N4 h3 [/ B: v2 ^: b
Exercises 232" {% g& [+ G# V$ e
7 Iterative solution of linear algebraic equations 235
" N- o' e. ~: p* G5 X) V7.1 Basic iterative schemes in explicit form 2372 P( E" C4 d! \' B. N9 ]
7.2 Matrix form of iteration methods and
# {  j* M! ?& |their convergence 2395 g  k( B) {/ o$ e2 `+ v/ v, W* r
7.3 Fourier analysis of convergence 244. b9 a7 N5 e. e4 \% R/ v( s
7.4 Application to an example 248
' c* y7 R5 s0 r/ i/ t4 V, d, j7.5 Extensions and related iterative methods 250" r: l/ ^  A6 q- f
7.6 The multigrid method 252
, d9 U# Q; r4 Z; ~8 B- ]4 T, Y7.7 The conjugate gradient method 258! }7 f0 D* _4 l9 ?- _
7.8 A numerical example: comparisons 261# a* M7 x0 n( R: V
Bibliographic notes 2635 R; L$ D- Q' k  a1 X* R
Exercises 263! }6 o' \% ?7 Q+ U
References 267
, D6 t2 i0 ~- l2 o: WIndex 273