RUNGE-KUTTA RESEARCH AT TORONTO

The main purpose of this paper is to review the work on Runge-Kutta me thods at the University of Toronto during the period 1963 to the prese nt (1996). To provide some background, brief mention is also made of r elated work on the numerical solution of ordinary differential equatio ns, but, with just a few exceptions, specific references are given onl y if the referenced material has a direct bearing on Runge-Kutta metho ds and their application to a variety of problem areas. There are seve ral main themes. New Runge-Kutta formulas and new error control strate gies are developed, leading for example to continuous methods and thei r application to areas such as delay, differential-algebraic and bound ary-value problems. Software design and implementation are also emphas ized. And so is the importance of cartful testing and comparing. Other topics, such as the notion of effectiveness, taking advantage of para llelism, and handling discontinuities, are also discussed.