剑桥出版社---偏微分方程数值解法 - 数学建模社区-数学中国

原版英文书第二版
1 F1 N4 \0 M9 M* ~3 @; N# ^( kcontents:+ g  m3 D& Z" B' i. }  m( R  d
Preface to the first edition page viii
# ]( v) O& h! L+ v) K( BPreface to the second edition xi
) P2 G$ @/ u+ I1 Introduction 1# m) ]5 B: T# B( D1 ]+ Y
2 Parabolic equations in one space variable 7
% M6 x2 M8 Z* t+ k' Y/ N8 U& k3 R2.1 Introduction 7
5 M; v- b7 i4 P( z: ~, b) h2.2 A model problem 7
4 e  Z( `  R& f- y$ C# t7 i" c2.3 Series approximation 9: W! d0 X3 O" o2 b
2.4 An explicit scheme for the model problem 10
  _* I' V# v; _) v3 o2.5 Difference notation and truncation error 12- P4 [' t6 S. M4 r' ?
2.6 Convergence of the explicit scheme 16
6 i- `5 e2 I& ]/ }2.7 Fourier analysis of the error 19
6 a( e. q4 U5 w1 F8 S$ f& x; p2.8 An implicit method 22# d$ v* O! d+ f; @" O1 P
2.9 The Thomas algorithm 24
& Y1 B% e& C  `7 Q: d3 a7 a( N2.10 The weighted average or θ-method 26
1 y2 i8 ]  H2 E( t+ h2.11 A maximum principle and convergence- O  O. d4 Z; X+ {5 U
for μ(1−θ)≤ 1
; `; K% e! ~+ n3 b' Q! M7 |2 33
2 a: g; {! G- }2.12 A three-time-level scheme 38
8 z" Y5 }$ U+ V, B2.13 More general boundary conditions 39% Q7 ]! V% ^4 w  j
2.14 Heat conservation properties 44! ?  O0 d$ W- Y1 M$ R
2.15 More general linear problems 46
6 ~+ {% X+ V1 i8 C3 w, H2.16 Polar co-ordinates 52" N# m) {" F) O7 Z
2.17 Nonlinear problems 54* N1 n& U( I( @) _
Bibliographic notes 56
2 n# _7 Y; W7 U1 uExercises 56* s9 z  V" ]" b4 W6 r7 P# l! A% l
v
( ?+ K! K" M; o: F  Dvi Contents) c0 p6 j# Y! F$ o0 d/ h. M
3 2-D and 3-D parabolic equations 62
* E5 i1 U6 v9 w3.1 The explicit method in a rectilinear box 627 m6 v+ M: P7 v/ c2 \1 d( [
3.2 An ADI method in two dimensions 64
- [+ o+ O: J: z' P3.3 ADI and LOD methods in three dimensions 70
$ f* v" c: ?+ y: u* W4 F. Z+ T3.4 Curved boundaries 718 K8 l& U* Y+ B& X* ?( Y
3.5 Application to general parabolic problems 80
6 d4 I  K! b+ R9 [1 k) GBibliographic notes 83
' p3 i9 B; u2 l+ bExercises 83& ~& r  X. G0 s
4 Hyperbolic equations in one space dimension 86
0 Z4 _- K5 j6 n9 x7 g3 S5 M4.1 Characteristics 86( r; e5 O2 |; W
4.2 The CFL condition 899 n8 h5 O6 `! i
4.3 Error analysis of the upwind scheme 94
9 U! N* M; U- |* i8 n1 ~" ^- N4.4 Fourier analysis of the upwind scheme 97
' |" n. y3 K* \! N, Z! H4.5 The Lax–Wendroff scheme 100
2 d4 C# ]+ ~+ t% P4.6 The Lax–Wendroff method for conservation laws 103
# U# ~$ y2 y4 D' H& W0 Y. A4.7 Finite volume schemes 110" \7 v. e8 b! r/ H& a. o* O
4.8 The box scheme 116
/ c4 v# k! H- z" T4.9 The leap-frog scheme 123
9 [; T$ G: F! O% ^3 C( p4.10 Hamiltonian systems and symplectic
: N$ U( K+ w5 P, Y% g- e" _  `9 V. {integration schemes 128
9 y/ x0 F  n4 U$ n: F% e: Y: T+ |/ f4.11 Comparison of phase and amplitude errors 135$ l' |8 B! d, j' [) t
4.12 Boundary conditions and conservation properties 139
& Q$ H1 k- y3 h" ~4.13 Extensions to more space dimensions 1438 Q% S7 o! w) [, i
Bibliographic notes 1465 q; L+ c& a7 k
Exercises 146' c( a  p; _- J) X5 Z( n8 o
5 Consistency, convergence and stability 151, o; c0 p$ w5 {
5.1 Definition of the problems considered 151
/ D2 o1 N. w' U+ [1 f9 f5.2 The finite difference mesh and norms 1520 j4 ^) D0 o$ A$ }; ^3 h
5.3 Finite difference approximations 154
6 ~" Z5 e) g* y) [6 v5.4 Consistency, order of accuracy and convergence 156
5 b# a. j* x( b- E3 ^) a5.5 Stability and the Lax Equivalence Theorem 157
' G* M6 j. i) m$ Y0 C5.6 Calculating stability conditions 1604 V+ R1 z3 G+ W1 @9 P: U. ?5 R0 a
5.7 Practical (strict or strong) stability 166
9 r+ e( ~% @5 }8 n, p- S1 ~, n5.8 Modified equation analysis 169
/ I( T6 N( ?. @5 r5 y5.9 Conservation laws and the energy method of analysis 177" D  o4 T3 u, C6 N1 i
5.10 Summary of the theory 186
; D4 q: K; g5 N& ^, L" G$ h( P  KBibliographic notes 189
; K6 b  [7 G7 T. ^/ T. F# @Exercises 1904 u. V3 I& ^( c! H5 F
Contents vii
3 {' i. L, x0 s% I! o/ W! [: a4 o6 Linear second order elliptic equations in
. t9 `! I  U, t1 X8 W: M; M3 t/ Rtwo dimensions 194  ]5 B  w& a: d& ]! a& p' Y
6.1 A model problem 1948 c. E& b8 O8 w1 B- f' d8 \) U/ S
6.2 Error analysis of the model problem 195
: u( E( @$ ~1 U* G$ `: o" i; s# T6.3 The general diffusion equation 197
7 F0 u! q. l% J6.4 Boundary conditions on a curved boundary 199
! r+ u4 z) [. W0 {# U; n- z6.5 Error analysis using a maximum principle 203
) z3 U7 v1 y; x( {6.6 Asymptotic error estimates 2137 M9 w. s- a( H- O+ [0 H+ \* _
6.7 Variational formulation and the finite- i8 y( B# X2 d2 B2 H* x& x: c" P
element method 218
0 R) F# V1 j  _. ~5 d6.8 Convection–diffusion problems 224
9 l/ [  A4 X& G( D) j' ]6.9 An example 2284 k1 e6 k/ y; p
Bibliographic notes 2316 j6 v1 K% h* ^+ e0 Z1 N
Exercises 232
; f, t  e3 U# ^3 W6 |* m. D2 Y7 Iterative solution of linear algebraic equations 235
/ c+ m* ^: k% n7.1 Basic iterative schemes in explicit form 237
( Z1 M+ S5 J1 ~) A/ s9 n, D7.2 Matrix form of iteration methods and# r4 |+ l; p( T$ B( A; m7 ]
their convergence 239# I$ w" M1 s* p0 g. W$ A% u  J# s
7.3 Fourier analysis of convergence 244. u  J3 V: Z6 `- V) l+ J+ {% t
7.4 Application to an example 248
- o, m4 H: G# a& M7.5 Extensions and related iterative methods 250- @, g2 @: r0 d. L) m: ]( Q- z6 N; y
7.6 The multigrid method 252
7.7 The conjugate gradient method 258! q# S4 s# g2 {. B; v6 }
7.8 A numerical example: comparisons 261/ `$ I9 ]% b% _) f
Bibliographic notes 263
( X/ K- F  b2 b% m4 }/ g  I# g: aExercises 263
- D0 d9 k6 z" y+ RReferences 267
* o$ T* Q* K- _3 j- nIndex 273