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Cambridge University Press
A First Course in the Numerical Analysis of Differential Equations / Edition 1

A First Course in the Numerical Analysis of Differential Equations / Edition 1

by Arieh Iserles


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A First Course in the Numerical Analysis of Differential Equations / Edition 1

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.

Product Details

ISBN-13: 9780521556552
Publisher: Cambridge University Press
Publication date: 01/28/1996
Series: Texts in Applied Mathematics Series
Edition description: Older Edition
Pages: 378
Product dimensions: 6.85(w) x 9.72(h) x 0.94(d)

About the Author

Arieh Iserles is a Professor in Numerical Analysis of Differential Equations in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge. He has been awarded the Onsager medal and served as a chair of the Society for Foundations of Computational Mathematics. He is also Managing Editor of Acta Numerica, Editor in Chief of Foundations of Computational Mathematics, and an editor of numerous other publications.

Table of Contents

Preface to the second edition ix

Preface to the first edition xiii

Flowchart of contents xix

I Ordinary differential equations 1

1 Euler's method and beyond 3

1.1 Ordinary differential equations and the Lipschitz condition 3

1.2 Euler's method 4

1.3 The trapezoidal rule 8

1.4 The theta method 13

Comments and bibliography 15

Exercises 16

2 Multistep methods 19

2.1 The Adams method 19

2.2 Order and convergence of multistep methods 21

2.3 Backward differentiation formulae 26

Comments and bibliography 28

Exercises 31

3 Runge-Kutta methods 33

3.1 Gaussian quadrature 33

3.2 Explicit Runge-Kutta schemes 38

3.3 Implicit Runge-Kutta schemes 41

3.4 Collocation and IRK methods 43

Comments and bibliography 48

Exercises 50

4 Stiff equations 53

4.1 What are stiff ODEs? 53

4.2 The linear stability domain and A-stability 56

4.3 A-stability of Runge-Kutta methods 59

4.4 A-stability of multistep methods 63

Comments and bibliography 68

Exercises 70

5 Geometric numerical integration 73

5.1 Between quality and quantity 73

5.2 Monotone equations and algebraic stability 77

5.3 From quadratic invariants to orthogonal flows 83

5.4 Hamiltonian systems 87

Comments and bibliography 95

Exercises 99

6 Error control 105

6.1 Numerical software vs. numerical mathematics 105

6.2 The Milne device 107

6.3 Embedded Runge-Kutta methods 113

Comments and bibliography 119

Exercises 121

7 Nonlinear algebraic systems 123

7.1 Functional iteration 123

7.2 The Newton-Raphson algorithm and its modification 127

7.3 Starting and stopping the iteration 130

Comments and bibliography 132

Exercises 133

II The Poisson equation 137

8 Finitedifference schemes 139

8.1 Finite differences 139

8.2 The five-point formula for ∇2u = f 147

8.3 Higher-order methods for ∇2u = f 158

Comments and bibliography 163

Exercises 166

9 The finite element method 171

9.1 Two-point boundary value problems 171

9.2 A synopsis of FEM theory 184

9.3 The Poisson equation 192

Comments and bibliography 200

Exercises 201

10 Spectral methods 205

10.1 Sparse matrices vs. small matrices 205

10.2 The algebra of Fourier expansions 211

10.3 The fast Fourier transform 214

10.4 Second-order elliptic PDEs 219

10.5 Chebyshev methods 222

Comments and bibliography 225

Exercises 230

11 Gaussian elimination for sparse linear equations 233

11.1 Banded systems 233

11.2 Graphs of matrices and perfect Cholesky factorization 238

Comments and bibliography 243

Exercises 246

12 Classical iterative methods for sparse linear equations 251

12.1 Linear one-step stationary schemes 251

12.2 Classical iterative methods 259

12.3 Convergence of successive over-relaxation 270

12.4 The Poisson equation 281

Comments and bibliography 286

Exercises 288

13 Multigrid techniques 291

13.1 In lieu of a justification 291

13.2 The basic multigrid technique 298

13.3 The full multigrid technique 302

13.4 Poisson by multigrid 303

Comments and bibliography 307

Exercises 308

14 Conjugate gradients 309

14.1 Steepest, but slow, descent 309

14.2 The method of conjugate gradients 312

14.3 Krylov subspaces and preconditioners 317

14.4 Poisson by conjugate gradients 323

Comments and bibliography 325

Exercises 327

15 Fast Poisson solvers 331

15.1 TST matrices and the Hockney method 331

15.2 Fast Poisson solver in a disc 336

Comments and bibliography 342

Exercises 344

III Partial differential equations of evolution 347

16 The diffusion equation 349

16.1 A simple numerical method 349

16.2 Order, stability and convergence 355

16.3 Numerical schemes for the diffusion equation 362

16.4 Stability analysis I: Eigenvalue techniques 368

16.5 Stability analysis II: Fourier techniques 372

16.6 Splitting 378

Comments and bibliography 381

Exercises 383

17 Hyperbolic equations 387

17.1 Why the advection equation? 387

17.2 Finite differences for the advection equation 394

17.3 The energy method 403

17.4 The wave equation 407

17.5 The Burgers equation 413

Comments and bibliography 418

Exercises 422

Appendix Bluffer's guide to useful mathematics 427

A.1 Linear algebra 428

A.1.1 Vector spaces 428

A.1.2 Matrices 429

A.1.3 Inner products and norms 432

A.1.4 Linear systems 434

A.1.5 Eigenvalues and eigenvectors 437

Bibliography 439

A.2 Analysis 439

A.2.1 Introduction to functional analysis 439

A.2.2 Approximation theory 442

A.2.3 Ordinary differential equations 445

Bibliography 446

Index 447

What People are Saying About This

From the Publisher

'A well written and exciting book … the exposition throughout is clear and very lively. The author's enthusiasm and wit are obvious on almost every page and I recommend the text very strongly indeed.' Proceedings of the Edinburgh Mathematical Society

'This is a well-written, challenging introductory text that addresses the essential issues in the development of effective numerical schemes for the solution of differential equations: stability, convergence, and efficiency. The soft cover edition is a terrific buy - I highly recommend it.' Mathematics of Computation

'This book can be highly recommended as a basis for courses in numerical analysis.' Zentralblatt fur Mathematik

'The overall structure and the clarity of the exposition make this book an excellent introductory textbook for mathematics students. It seems also useful for engineers and scientists who have a practical knowledge of numerical methods and wish to acquire a better understanding of the subject.' Mathematical Reviews

'… nicely crafted and full of interesting details.' ITW Nieuws

'I believe this book succeeds. It provides an excellent introduction to the numerical analysis of differential equations . . .' Computing Reviews

'As a mathematician who developed an interest in numerical analysis in the middle of his professional career, I thoroughly enjoyed reading this text. I wish this book had been available when I first began to take a serious interest in computation. The author's style is comfortable . . . This book would be my choice for a text to 'modernize' such courses and bring them closer to the current practice of applied mathematics.' American Journal of Physics

'Iserles has successfully presented, in a mathematically honest way, all essential topics on numerical methods for differential equations, suitable for advanced undergraduate-level mathematics students.' Georgios Akrivis, University of Ioannina, Greece

'The present book can, because of the extension even more than the first edition, be highly recommended for readers from all fields, including students and engineers.' Zentralblatt MATH

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