A first course in the numerical analysis of differential equations

TLDR

Numerical analysis is viewed differently by mathematicians, scientists, engineers, and computer scientists, yet this book unifies these perspectives. The book aims to reconcile these viewpoints by providing a rigorous account of numerical analysis for ordinary and partial differential equations. It covers multistep and Runge–Kutta schemes, finite difference and finite element methods for Poisson, and algorithms for large sparse linear systems. The updated edition adds chapters on geometric numerical integration, spectral methods, and conjugate gradients.

Abstract

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.