A first course in the numerical analysis of differential equations

Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface to the second edition......Page 11
Preface to the first edition......Page 15
Flowchart of contents......Page 21
PART I Ordinary differential equations......Page 23
1.1 Ordinary differential equations and the Lipschitz condition......Page 25
1.2 Euler’s method......Page 26
1.3 The trapezoidal rule......Page 30
1.4 The theta method......Page 35
Comments and bibliography......Page 37
Exercises......Page 38
2.1 The Adams method......Page 41
2.2 Order and convergence of multistep methods......Page 43
2.3 Backward differentiation formulae......Page 48
Comments and bibliography......Page 50
Exercises......Page 53
3.1 Gaussian quadrature......Page 55
3.2 Explicit Runge–Kutta schemes......Page 60
3.3 Implicit Runge–Kutta schemes......Page 63
3.4 Collocation and IRK methods......Page 65
Comments and bibliography......Page 70
Exercises......Page 72
4.1 What are stiff? ODEs?......Page 75
4.2 The linear stability domain and A-stability......Page 78
4.3 A-stability of Runge–Kutta methods......Page 81
4.4 A-stability of multistep methods......Page 85
Comments and bibliography......Page 90
Exercises......Page 92
5.1 Between quality and quantity......Page 95
5.2 Monotone equations and algebraic stability......Page 99
5.3 From quadratic invariants to orthogonal flows......Page 105
5.4 Hamiltonian systems......Page 109
Comments and bibliography......Page 117
Exercises......Page 121
6.1 Numerical software vs. numerical mathematics......Page 127
6.2 The Milne device......Page 129
6.3 Embedded Runge–Kutta methods......Page 135
Comments and bibliography......Page 141
Exercises......Page 143
7.1 Functional iteration......Page 145
7.2 The Newton–Raphson algorithm and its modification......Page 149
7.3 Starting and stopping the iteration......Page 152
Comments and bibliography......Page 154
Exercises......Page 155
PART II The Poisson equation......Page 159
8.1 Finite differences......Page 161
8.2 The five-point formula for…......Page 169
8.3 Higher-order methods for…......Page 180
Comments and bibliography......Page 185
Exercises......Page 188
9.1 Two-point boundary value problems......Page 193
9.2 A synopsis of FEM theory......Page 206
9.3 The Poisson equation......Page 214
Comments and bibliography......Page 222
Exercises......Page 223
10.1 Sparse matrices vs. small matrices......Page 227
10.2 The algebra of Fourier expansions......Page 233
10.3 The fast Fourier transform......Page 236
10.4 Second-order elliptic PDEs......Page 241
10.5 Chebyshev methods......Page 244
Comments and bibliography......Page 247
Exercises......Page 252
11.1 Banded systems......Page 255
11.2 Graphs of matrices and perfect Cholesky factorization......Page 260
Comments and bibliography......Page 265
Exercises......Page 268
12.1 Linear one-step stationary schemes......Page 273
12.2 Classical iterative methods......Page 281
12.3 Convergence of successive over-relaxation......Page 292
12.4 The Poisson equation......Page 303
Comments and bibliography......Page 308
Exercises......Page 310
13.1 In lieu of a justification......Page 313
13.2 The basic multigrid technique......Page 320
13.3 The full multigrid technique......Page 324
13.4 Poisson by multigrid......Page 325
Comments and bibliography......Page 329
Exercises......Page 330
14.1 Steepest, but slow, descent......Page 331
14.2 The method of conjugate gradients......Page 334
14.3 Krylov subspaces and preconditioners......Page 339
14.4 Poisson by conjugate gradients......Page 345
Comments and bibliography......Page 347
Exercises......Page 349
15.1 TST matrices and the Hockney method......Page 353
15.2 Fast Poisson solver in a disc......Page 358
Comments and bibliography......Page 364
Exercises......Page 366
PART III Partial differential equations of evolution......Page 369
16.1 A simple numerical method......Page 371
16.2 Order, stability and convergence......Page 377
16.3 Numerical schemes for the diffiusion equation......Page 384
16.4 Stability analysis I: Eigenvalue techniques......Page 390
16.5 Stability analysis II: Fourier techniques......Page 394
16.6 Splitting......Page 400
Comments and bibliography......Page 403
Exercises......Page 405
17.1 Why the advection equation?......Page 409
17.2 Finite di?erences for the advection equation......Page 416
17.3 The energy method......Page 425
17.4 The wave equation......Page 429
17.5 The Burgers equation......Page 435
Comments and bibliography......Page 440
Exercises......Page 444
Appendix Bluffer’s guide to useful mathematics......Page 449
A.1.1 Vector spaces......Page 450
A.1.2 Matrices......Page 451
A.1.3 Inner products and norms......Page 454
A.1.4 Linear systems......Page 456
A.1.5 Eigenvalues and eigenvectors......Page 459
A.2.1 Introduction to functional analysis......Page 461
A.2.2 Approximation theory......Page 464
A.2.3 Ordinary differential equations......Page 467
Bibliography......Page 468
Index......Page 469