A First Course in the Numerical Analysis of Differential Equations, 2nd Edition

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 exposition maintains a balance between theoretical, algorithmic and applied aspects. This new edition has been extensively updated, and includes new chapters on emerging subject areas: geometric numerical integration, spectral methods and conjugate gradients. Other topics covered include multistep and Runge-Kutta methods; finite difference and finite elements techniques for the Poisson equation; and a variety of algorithms to solve large, sparse algebraic systems.

Table of Contents

PART I Ordinary differential equations
1 Euler’s method and beyond
2 Multistep methods
3 Runge–Kutta methods
4 Stiff equations
5 Geometric numerical integration
6 Error control
7 Nonlinear algebraic systems

PART II The Poisson equation
8 Finite difference schemes
9 The finite element method
10 Spectral methods
11 Gaussian elimination for sparse linear equations
12 Classical iterative methods for sparse linear equations
13 Multigrid techniques
14 Conjugate gradients
15 Fast Poisson solvers
16 The difiusion equation
17 Hyperbolic equations

Appendix Bluffer’s guide to useful mathematics
A.1 Linear algebra
A.2 Analysis