Abstract

John Butcher originally took up Numerical Analysis as a hobby after hearing a talk by Merson in 1957 while he was completing a Ph.D in Physics on Cosmic Rays. Whether this was a great loss to Physics it is of course impossible to determine but it was certainly of enormous benefit to Numerical Analysis and the field of the numerical solution of ordinary differential equations in particular. Since the publication of his first paper (in ordinary differential equations) in 1963 entitled "Coefficients for the study of Runge-Kutta integration processes", John Butcher has devoted half of his life to this "hobby". His work has always been innovative and lucid, characterized by considerable mathematical technique and a clarity of presentation. This paper attempts to trace his work not only in a technical sense but in terms of the development of certain themes which have kept recurring in his work over the last thirty years. In addition, this paper also attempts to relate his work to many of the important developments which have taken place in this field in the last thirty-five years. 0 Preamble There have been many major contributions to the field of numerical methods for ordinary differential equations of initial value type (IVPs), but perhaps the two most significant contributors in the last forty years have been Germund Dahlquist and John Butcher. Although they worked in entirely different areas (Dahlquist on linear multistep methods and Butcher on multi-stage methods), much of their work, especially in nonlinear stability theory parallels one another. Indeed some of Butcher's work on the B-stability of Runge-Kutta methods [1975] and the equivalence of AN-stability and BN-stability for multivalue methods [1987, 1987a] owe much to the work of Dahlquist on the s...

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