A First Course in the Numerical Analysis of Differential Equations

by Arieh Iserles

• ISBN13: 9780521734905
• ISBN10: 0521734908
• Edition: 2nd
• Format: Paperback
• Copyright: 2008-12-29
• Publisher: Cambridge University Press

Summary

Numerical analysis presents different faces to the world. For mathematicians it is a bona fide mathematical theory with an applicable flavour. For scientists and engineers it is a practical, applied subject, part of the standard repertoire of modelling techniques. For computer scientists it is a theory on the interplay of computer architecture and algorithms for real-number calculations. The tension between these standpoints is the driving force of this book, which presents a rigorous account of the fundamentals of numerical analysis of both ordinary and partial differential equations. The point of departure is mathematical but the exposition strives to maintain a balance between theoretical, algorithmic and applied aspects of the subject. In detail, topics covered include numerical solution of ordinary differential equations by multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; a variety of algorithms to solve large, sparse algebraic systems; methods for parabolic and hyperbolic differential equations and techniques of their analysis. The book is accompanied by an appendix that presents brief back-up in a number of mathematical topics. Dr Iserles concentrates on fundamentals: deriving methods from first principles, analysing them with a variety of mathematical techniques and occasionally discussing questions of implementation and applications. By doing so, he is able to lead the reader to theoretical understanding of the subject without neglecting its practical aspects. The outcome is a textbook that is mathematically honest and rigorous and provides its target audience with a wide range of skills in both ordinary and partial differential equations.

Table of Contents

Preface to the second edition	p. ix
Preface to the first edition	p. xiii
Flowchart of contents	p. xix
Ordinary differential equations	p. 1
Euler's method and beyond	p. 3
Ordinary differential equations and the Lipschitz condition	p. 3
Euler's method	p. 4
The trapezoidal rule	p. 8
The theta method	p. 13
Comments and bibliography	p. 15
Exercises	p. 16
Multistep methods	p. 19
The Adams method	p. 19
Order and convergence of multistep methods	p. 21
Backward differentiation formulae	p. 26
Comments and bibliography	p. 28
Exercises	p. 31
Runge-Kutta methods	p. 33
Gaussian quadrature	p. 33
Explicit Runge-Kutta schemes	p. 38
Implicit Runge-Kutta schemes	p. 41
Collocation and IRK methods	p. 43
Comments and bibliography	p. 48
Exercises	p. 50
Stiff equations	p. 53
What are stiff ODEs?	p. 53
The linear stability domain and A-stability	p. 56
A-stability of Runge-Kutta methods	p. 59
A-stability of multistep methods	p. 63
Comments and bibliography	p. 68
Exercises	p. 70
Geometric numerical integration	p. 73
Between quality and quantity	p. 73
Monotone equations and algebraic stability	p. 77
From quadratic invariants to orthogonal flows	p. 83
Hamiltonian systems	p. 87
Comments and bibliography	p. 95
Exercises	p. 99
Error control	p. 105
Numerical software vs. numerical mathematics	p. 105
The Milne device	p. 107
Embedded Runge-Kutta methods	p. 113
Comments and bibliography	p. 119
Exercises	p. 121
Nonlinear algebraic systems	p. 123
Functional iteration	p. 123
The Newton-Raphson algorithm and its modification	p. 127
Starting and stopping the iteration	p. 130
Comments and bibliography	p. 132
Exercises	p. 133
The Poisson equation	p. 137
Finite difference schemes	p. 139
Finite differences	p. 139
The five-point formula for ∇2u = f	p. 147
Higher-order methods for ∇2u = f	p. 158
Comments and bibliography	p. 163
Exercises	p. 166
The finite element method	p. 171
Two-point boundary value problems	p. 171
A synopsis of FEM theory	p. 184
The Poisson equation	p. 192
Comments and bibliography	p. 200
Exercises	p. 201
Spectral methods	p. 205
Sparse matrices vs. small matrices	p. 205
The algebra of Fourier expansions	p. 211
The fast Fourier transform	p. 214
Second-order elliptic PDEs	p. 219
Chebyshev methods	p. 222
Comments and bibliography	p. 225
Exercises	p. 230
Gaussian elimination for sparse linear equations	p. 233
Banded systems	p. 233
Graphs of matrices and perfect Cholesky factorization	p. 238
Comments and bibliography	p. 243
Exercises	p. 246
Classical iterative methods for sparse linear equations	p. 251
Linear one-step stationary schemes	p. 251
Classical iterative methods	p. 259
Convergence of successive over-relaxation	p. 270
The Poisson equation	p. 281
Comments and bibliography	p. 286
Exercises	p. 288
Multigrid techniques	p. 291
In lieu of a justification	p. 291
The basic multigrid technique	p. 298
The full multigrid technique	p. 302
Poisson by multigrid	p. 303
Comments and bibliography	p. 307
Exercises	p. 308
Conjugate gradients	p. 309
Steepest, but slow, descent	p. 309
The method of conjugate gradients	p. 312
Krylov subspaces and preconditioners	p. 317
Poisson by conjugate gradients	p. 323
Comments and bibliography	p. 325
Exercises	p. 327
Fast Poisson solvers	p. 331
TST matrices and the Hockney method	p. 331
Fast Poisson solver in a disc	p. 336
Comments and bibliography	p. 342
Exercises	p. 344
Partial differential equations of evolution	p. 347
The diffusion equation	p. 349
A simple numerical method	p. 349
Order, stability and convergence	p. 355
Numerical schemes for the diffusion equation	p. 362
Stability analysis I: Eigenvalue techniques	p. 368
Stability analysis II: Fourier techniques	p. 372
Splitting	p. 378
Comments and bibliography	p. 381
Exercises	p. 383
Hyperbolic equations	p. 387
Why the advection equation?	p. 387
Finite differences for the advection equation	p. 394
The energy method	p. 403
The wave equation	p. 407
The Burgers equation	p. 413
Comments and bibliography	p. 418
Exercises	p. 422
Appendix Bluffer's guide to useful mathematics	p. 427
Linear algebra	p. 428
Vector spaces	p. 428
Matrices	p. 429
Inner products and norms	p. 432
Linear systems	p. 434
Eigenvalues and eigenvectors	p. 437
Bibliography	p. 439
Analysis	p. 439
Introduction to functional analysis	p. 439
Approximation theory	p. 442
Ordinary differential equations	p. 445
Bibliography	p. 446
Index	p. 447
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