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.