Numerical Solution of Partial Differential Equations: Finite Difference Methods

Smith G. D. Numerical Solution of Partial Differential Equations 3rd ......Page 2
Copyright ......Page 5
Preface to the third edition ......Page 6
Contents ......Page 8
NOTATION xiii ......Page 13
Descriptive treatment of elliptic equations 1 ......Page 14
Descriptive treatment of parabolic and hyperbolic equations 4 ......Page 17
Finite-difference approximations to derivatives 6 ......Page 19
Notation for functions of several variables 8 ......Page 21
Transformation to non-dimensional form 11 ......Page 24
An explicit finite-difference approximation to dU/dt = d2U/dx2 12 ......Page 25
A worked example covering three cases and including comparison tables 13 ......Page 26
Crank-Nicolson implicit method 19 ......Page 32
Worked example including a comparison table 21 ......Page 34
Solution of the implicit equations by Gauss’s elimination method 24 ......Page 37
The stability of the elimination method 27 ......Page 40
A weighted average approximation 28 ......Page 41
Derivative boundary conditions 29 ......Page 42
ii) Explicit formula and forward-differenced boundary condition 33 ......Page 44
iii) Implicit formula and central-differenced boundary condition 36 ......Page 49
The local truncation error and a worked example 38 ......Page 51
Consistency and a worked example illustrating both consistency and inconsistency 40 ......Page 53
Convergence, descriptive treatment and the analysis of an explicit approximation 43 ......Page 56
Stability, descriptive treatment 47 ......Page 60
Vector and matrix norms, subordinate matrix norms, p(A)^ UAH 49 ......Page 62
A necessary and sufficient condition for stability, jjAH^l, and two worked examples 51 ......Page 64
Matrix method of analysis, fixed mesh size. 57 ......Page 70
A note on the eigenvalues of /(A) and [/!(A)]-1/2(A) 58 ......Page 71
The eigenvalues of a common tridiagonal matrix 59 ......Page 72
Theorems on bounds for eigenvalues and an application.(Gerschgorin’s theorems) 60 ......Page 73
Gerschgorin’s circle theorem and the norm of matrix A 62 ......Page 75
Stability criteria for derivative boundary conditions using (i) the circle theorem (ii) IjAjloo^l 63 ......Page 76
Stability condition allowing exponential growth 66 ......Page 79
Stability, von Neumann’s method, and three worked examples 67 ......Page 80
The global rounding error 71 ......Page 84
Lax’s equivalence theorem (statement only) and a detailed analysis of a simple case 72 ......Page 85
Finite-difference approximations to dU/dt = V2U in cylindrical and spherical polar co-ordinates 75 ......Page 88
A worked example involving lim(dU/dx)/x 77 ......Page 90
Exercises and solutions 79 ......Page 92
Reduction to a system of ordinary differential equations 111 ......Page 124
A note on the solution of dV/df = AV + b 113 ......Page 126
Finite-difference approximations via the ordinary differential equations 115 ......Page 128
The Pade approximants to exp 6 116 ......Page 129
Standard finite-difference equations via the Pade approximants 117 ......Page 130
A0-stability, L0-stability and the symbol of the method 119 ......Page 132
A necessary constraint on the time step for the Crank- Nicolson method 122 ......Page 135
The local truncation errors associated with the Pade approximants 124 ......Page 137
An extrapolation method for improving accuracy in t 126 ......Page 139
The symbol for the extrapolation method 128 ......Page 141
The arithmetic of the extrapolation method 129 ......Page 142
i) Preliminary results 132 ......Page 145
ii) The eigenvalue-eigenvector solution of dV/dt = AV 134 ......Page 147
iii) An application giving an approximate solution for large t 135 ......Page 148
i) Reduction of the local truncation error-the Douglas equations 137 ......Page 149
ii) Use of three time-level difference equations 138 ......Page 150
iii) Deferred correction method 139 ......Page 151
iv) Richardson’s deferred approach to the limit 141 ......Page 154
i) Newton’s linearization method and a worked example 142 ......Page 155
ii) Richtmyer’s linearization method 144 ......Page 157
iii) Lee’s three time-level method 146 ......Page 159
A comparison of results for methods (i), (ii), and (iii) for a particular problem 147 ......Page 160
i) A useful theorem on eigenvalues 148 ......Page 161
iii) A worked example 150 ......Page 163
Introduction to the analytical solution of homogeneous difference equations: 153 ......Page 166
i) The eigenvalues and vectors of a common tridiagonal matrix 154 ......Page 167
ii) The analytical solution of the classical explicit approximation to dU/dt = d2U/dx2 156 ......Page 169
Exercises and solutions 158 ......Page 171
Analytical solution of first-order quasi-linear equations 175 ......Page 188
A worked example and discussion 176 ......Page 189
Numerical integration along a characteristic 178 ......Page 191
A worked example 179 ......Page 192
i) Lax-Wendroff explicit method and a worked example with a comparison table 181 ......Page 194
ii) Lax-Wendroff method for a set of simultaneous equations 183 ......Page 196
iii)The Courant-Friedrichs-Lewy condition 186 ......Page 199
iv) Wendroff’s implicit approximation 187 ......Page 200
i) Discontinuous initial values 188 ......Page 201
ii) Discontinuous initial derivatives 189 ......Page 202
Discontinuities and finite-difference approximations. An example using Wendroff’s implicit approximation 190 ......Page 203
Reduction of a first-order equation to a system of ordinary differential equations 193 ......Page 206
The (1,0) Pade difference approximation 195 ......Page 208
The (1,1) Pade or Crank-Nicolson difference equations 196 ......Page 209
An improved approximation to dU/dx and the (1,0) Pade difference equations 197 ......Page 210
A word of caution on the central-difference approximation to dU/dx 200 ......Page 213
Second-order quasi-linear hyperbolic equations. Characteristic curves, and the differential relationship along them 202 ......Page 215
Numerical solution by the method of characteristics 204 ......Page 217
A worked example 207 ......Page 220
A characteristic as an initial curve 209 ......Page 222
Propagation of discontinuities, second-order equations 210 ......Page 223
i) Explicit methods and the Courant-Friedrichs-Lewy condition 213 ......Page 226
ii) Implicit methods with particular reference to the wave-equation 216 ......Page 229
Simultaneous first-order equations and stability 217 ......Page 230
Exercises and solutions 220 ......Page 233
Introduction 239 ......Page 252
Worked examples: (i) A torsion problem, (ii) A heat-conduction problem with derivative boundary conditions 240 ......Page 253
Finite-differences in polar co-ordinates 245 ......Page 258
Formulae for derivatives near a curved boundary 247......Page 260
Improvement of the accuracy of solutions: (i) Finer mesh, (ii) 248......Page 261
Deferred approach to the limit, (iii) Deferred correction method, (iv) More accurate finite-difference formulae including the nine-point formula 249 ......Page 262
Analysis of the discretization error of the five-point approximation to Poisson’s equation over a rectangle. Quoted 252 ......Page 265
result for irregular boundaries 254 ......Page 267
Comments on the solution of difference equations, covering Gauss elimination, LU decomposition, rounding errors, ill- conditioning, iterative refinement, iterative methods 257 ......Page 270
Systematic iterative methods for large linear systems 260 ......Page 273
Jacobi, Gauss-Seidel, and SOR methods 261 ......Page 274
A worked example covering each method 263 ......Page 276
Jacobi, Gauss-Seidel, and SOR methods in matrix form 266 ......Page 279
A necessary and sufficient condition for convergence of iterative methods 268 ......Page 281
A sufficient condition for convergence 269 ......Page 282
Asymptotic and average rates of convergence 270 ......Page 283
Methods for accelerating convergence, (i) Lyusternik’s method, (ii) Aitken’s method. An illustrative example 272 ......Page 285
Eigenvalues of the Jacobi and SOR iteration matrices and two worked examples 275 ......Page 288
The optimum acceleration parameter for the SOR method. A necessary theorem 277 ......Page 290
Proof of (A + co — 1)2 = Aco2/ll2 for block tridiagonal coefficient matrices 279 ......Page 292
Non-zero eigenvalues of the Jacobi iteration matrix 280 ......Page 293
Theoretical determination of the optimum relaxation parameter (ob 282 ......Page 295
The Gauss-Seidel iteration matrix H(l) 285 ......Page 298
Re-ordering of equations and unknowns 286 ......Page 299
Point iterative methods and re-orderings 287 ......Page 300
Introduction to 2-cyclic matrices and consistent ordering 288 ......Page 301
2-cyclic matrices 289 ......Page 302
Ordering vectors for 2-cyclic matrices 290 ......Page 303
Consistent ordering of a 2-cyclic matrix 292 ......Page 305
The ordering vector for a block tridiagonal matrix 294 ......Page 307
Additional comments on consistent ordering and the SOR method 297 ......Page 310
Consistent orderings associated with the five-point approximation to Poisson’s equation 298 ......Page 311
Stone’s strongly implicit iterative method 302 ......Page 315
A recent direct method 309 ......Page 322
Exercises and solutions 311 ......Page 324
INDEX 334......Page 347
cover......Page 1