This excellent reference book is the first to attempt to survey the full range of methods for the parallel solution of ordinary differential equations (ODEs). The main emphasis is on the initial value problem. Conventional methods for this problem do not lend themselves naturally to parallelization. This point is made quite clear; the author maintains an appropriate skeptical tone, being particularly careful to identify the pitfalls when comparing the efficiencies of parallel methods with their sequential counterparts. As one would expect from an expert in the field, there are several new results from Burrage’s own research, but these are not given undue prominence.

To make a contribution in parallel methods for ODEs, considerable background is needed. The first four chapters provide this background, starting with a chapter on parallel computing, followed by two excellent chapters on sequential numerical methods and the corresponding theory for ODE initial value problems, then a wide-ranging chapter on parallel linear algebra. The chapters on numerical methods for ODEs are novel for relying mainly on the modern theory of multivalue methods to infer the corresponding properties of Runge-Kutta and multistep methods. What may be the most important chapter comes next. It provides a careful survey of the many attempts to find parallelism in, or introduce it into, the standard sequential approaches to solving the initial value problem. This is followed by an “other approaches and problems” chapter, which considers operator splitting, boundary value problems, and many other topics. The last three chapters discuss waveform relaxation--first continuous, then discrete, and finally its implementation and testing using the respected code VODE as the underlying sequential ODE software.

The book finishes with a bibliography of over 500 entries, about half of which are from after 1990 and the vast majority of the remainder from after 1980. In the preface, the author identifies a readership ranging from advanced undergraduates to researchers in the field. Few undergraduates will be able to handle the sweep, depth, and quantity of material here. However, this text is a must for all researchers in the numerical ODE community and for all scientists and engineers involved in the parallel solution of evolutionary problems modeled by differential equations. I recommend it highly.