Numerical Methods For Partial Differential Equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, 2ND ED.......Page 1
Half-title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 8
Preface to second edition......Page 12
Preface to first edition......Page 14
1-0 Introduction......Page 16
1-1 Classification of physical problems......Page 18
1-2 Classification of equations......Page 20
1-3 Asymptotics......Page 25
1-4 Discrete methods......Page 29
1-5 Finite differences and computational molecules......Page 30
1-6 Finite difference operators......Page 34
1-7 Errors......Page 38
1-8 Stability and convergence......Page 43
1-9 Irregular boundaries......Page 45
1-10 Choice of discrete network......Page 48
1-11 Dimensionless forms......Page 49
References......Page 54
2-0 Introduction......Page 56
2-1 Simple explicit methods......Page 57
2-2 Fourier stability method......Page 62
2-3 Implicit methods......Page 64
2-4 An unconditionally unstable difference equation......Page 70
2-5 Matrix stability analysis......Page 71
2-6 Extension of matrix stability analysis......Page 74
2-7 Consistency, stability, and convergence......Page 76
2-8 Pure initial value problems......Page 77
2-9 Variable coefficients......Page 79
(a) Diffusion in circular cylindrical coordinates......Page 83
(c) Diffusion with spherical symmetry......Page 84
2-11 General concepts of error reduction......Page 85
2-12 Explicit methods for nonlinear problems......Page 88
2-13 An application of the explicit method......Page 92
(a) Application of the backward difference......Page 97
(b) Crank–Nicolson forms......Page 99
(c) Predictor–corrector methods......Page 100
2-15 Concluding remarks......Page 104
References......Page 105
3-0 Introduction......Page 107
3-1 Simple finite difference schemes......Page 109
3-2 Iterative methods......Page 113
3-3 Linear elliptic equations......Page 115
(a) Jacobi method......Page 118
(b) Gauss–Seidel method......Page 119
(c) Successive over-relaxation (SOR)......Page 121
3-5 Convergence of point iterative methods......Page 122
3-6 Rates of convergence......Page 129
3-7 Accelerations—successive over-relaxation (SOR)......Page 134
3-8 Extensions of SOR......Page 140
3-9 Qualitative examples of over-relaxation......Page 145
(a) Gradient method......Page 150
(b) Richardson\'s method......Page 151
(c) Semi-iterative methods......Page 154
3-11 Block iterative methods......Page 159
3-12 Alternating direction methods......Page 163
3-13 Summary of ADI results......Page 167
(a) Mildly nonlinear elliptic equations......Page 173
(c) Laminar flow of non-Newtonian fluids......Page 174
References......Page 176
4-0 Introduction......Page 180
4-1 The quasilinear system......Page 185
(a) Direct calculation of the primitive variables......Page 191
(b) Stress wave propagation......Page 192
4-3 Method of characteristics......Page 195
4-4 Constant states and simple waves......Page 200
4-5 Typical application of characteristics......Page 201
4-6 Explicit finite difference methods......Page 208
4-7 Overstabiiity......Page 212
4-8 Implicit methods for second-order equations......Page 214
4-9 Nonlinear examples......Page 216
4-10 Simultaneous first-order equations—explicit methods......Page 218
4-12 Hybrid methods for first-order equations......Page 224
4-13 Gas dynamics in one-space variable......Page 227
(a) Method of Courant et al. [28]......Page 229
(b) Lelevier\'s scheme......Page 230
(c) Conservation form and Lax–Wendroff schemes......Page 231
4-15 Lagrangian difference equations......Page 234
4-16 Hopscotch methods for conservation laws......Page 236
4-17 Explicit-implicit schemes for conservation laws......Page 239
References......Page 242
5-1 Singularities......Page 245
(a) Subtracting out the singularity......Page 248
(c) The method of Motz and Woods......Page 249
(d) Removal of singularity......Page 251
5-2 Shocks......Page 253
(a) Pseudoviscosity......Page 255
(b) Lax–Wendroff method......Page 256
5-3 Eigenvalue problems......Page 259
(a) The membrane eigenvalue problemt [Eqn (5-45)]......Page 261
(b) Some methods of computation......Page 262
(c) A nonlinear eigenvalue problem (Motz [48])......Page 265
(a) Generalizations of the elementary methods......Page 266
(b) Alternating direction methods......Page 267
5-5 Additional comments on elliptic equations......Page 270
(a) Nonlinear over-relaxation......Page 271
(b) Some alternative procedures......Page 274
(c) Some questions associated with numerical weather prediction......Page 275
(a) Characteristics in three independent variables......Page 277
(b) Finite difference methods......Page 278
(c) Singularities in solutions of nonlinear hyperbolic equations......Page 280
5-7 Mixed systems......Page 285
(a) Elasticity......Page 289
(b) Vibrations of a thin beam......Page 294
5-9 Fluid mechanics: the Navier–Stokes equations......Page 296
(a) Stream function—vorticity method......Page 298
(b) Primitive variable methods......Page 306
(c) Vector potential methods......Page 309
5-10 Introduction to Monte Carlo methods......Page 314
5-11 Method of lines......Page 317
5-12 Fast Fourier transform and applications......Page 319
5-13 Method of fractional steps......Page 322
References......Page 326
6-1 Weighted residual methods (WRM)......Page 335
(c) Least squares (Gauss–Legendre; cf. Hall [18], Sorenson [19])......Page 337
(f) General WRM......Page 338
(g) Stationary functional method (Rayleigh [29], Ritz [30])......Page 339
6-2 Orthogonal collocation......Page 340
6-3 Bubnov–Galerkin (B-G) method......Page 344
6-4 Remarks on completeness, convergence, and error bounds......Page 348
6-5 Nagumo\'s lemma and application......Page 354
6-6 Introduction to finite elements......Page 357
References......Page 363
Author Index......Page 366
Subject Index......Page 372
Computer Science and Applied Mathematics: A Series of Monographs and Textbooks......Page 381