An Appreciation By

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 beneet to Numerical Analysis and the eld of the numerical solution of ordinary diierential equations in particular. Since the publication of his rst paper (in ordinary diierential equations) in 1963 entitled \Coeecients 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 eld in the last thirty-ve years. 0 Preamble There have been many major contributions to the eld of numerical methods for ordinary diierential equations of initial value type (IVPs), but perhaps the two most signiicant contributors in the last forty years have been Germund Dahlquist and John Butcher. Although they worked in entirely diierent 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 stability of linear multistep methods for stii nonlinear problems 1975] and the equivalence of G-stability and A-stability for one-leg methods 1978]. Not only were the ideas encapsulated in these two papers of Dahlquist of great signiicance in the non-linear stability work of Butcher but so were some of Dahlquist's mathematical techniques which Butcher extended and used in very clever ways. So profound has been the innuence of Germund Dahlquist and John Butcher in the numerical solution of IVPs that more than once they have been addressed, respectively, as \Le Roi de linear multistep" and \Le Roi de Runge-Kutta". 1 This paper has as its theme the work and impact of John Butcher on numerical methods for …