This first book of a two-volume series on numerical fluid dynamics is concerned with finite difference methods for initial boundary-value problems. The intention is to make the field of numerical fluid dynamics accessible by emphasizing concepts along with the underlying theory rather than presenting a collection of recipes and by building on the classical methods to develop the numerical methods on the leading edge of research.

This book, which is essentially self-contained, is intended to be a text directed at first-year graduate students in engineering and the physical sciences and as a reference to practitioners in the field. It assumes a basic knowledge of partial differential equations. The proofs of some of the classical theorems requiring a greater knowledge of analysis are presented in appendices. This first volume is designed for a course in numerical methods for initial boundaryvalue problems. The numerical methods and techniques for their analysis contained in Volume I provide the foundation for Volume II, which deals exclusively with the equations governing fluid motion.

I am indebted to many friends and colleagues for their support and cooperation. The formost of these, taught me numerical analysis, was my advisor, and is my friend, Alexandre Chorin. This book is in no small way a reflection of his influence on my way of viewing numerical fluid dynamics.

The comparison of algorithms

As we have introduced and discussed the various possible schemes for solving Volterra and Fredholm equations of the second kind, we have made a number of general qualitative and quantitative comparisons of the methods, in terms of their cost (for a given discretisation size N), their likely accuracy (for a given N) and rate of convergence (as N increases), their stability and the ease with which usable error estimates can be provided for the method. It is unfortunately a truism that the performance of a method in practice is also affected by apparently minor details of the implementation (that is, of the program). It is also true that comparisons based on one or two numerical examples can be misleading; an overall picture of the performance of a method depends on its behaviour over a wide class of problems. In this chapter we consider in some detail how objective comparisons between methods can be made and we illustrate the problems by looking at the performance of several specific routines for solving second kind Fredholm equations. These detailed comparisons are very interesting but, when two different algorithms give fairly close results, the reader is warned of the dangers of mentally ranking the algorithms: different implementations of the algorithms might well rank in the reverse order, so that only quite large differences in performance should be taken seriously.

Implementations of a given algorithm can be of two main types, namely automatic or non-automatic routines.

This book considers the practical solution of one-dimensional integral equations. Both integral equations, and methods for solving them, come in many forms and we could not try, and have not tried, to be exhaustive. For the problem classes covered, we have used the ‘classical’ Fredholm/Volterra/first kind/second kind/third kind categorisation. Not all problems fit neatly into such categories; then the methods used to solve standard classes of problems must be modified and tailored to suit the needs of nonstandard ‘real life’ problems. It is hoped that the nature of any such modifications will be obvious to the intelligent reader. Not all categories of problems seem equally important (i.e. frequent) in practice; we have tried to spend most time on the most important classes of problems.

We have also been selective in the choice of methods covered. Here, personal likes and dislikes have helped the selection process, but we have also taken particular note of the fact that the cost of solving even a one-dimensional integral equation of Fredholm type can be unexpectedly high. Methods which converge slowly but steadily are therefore not very attractive in practice and particular emphasis is placed on the ability of a given method to obtain rapid convergence, to provide computable error estimates and to produce reliable results at relatively low cost.

It is hoped that the book will serve as a reference text for the practising numerical mathematician, scientist or engineer, who finds integral problems arising in his work.