Runge-Kutta research at Toronto Original Research Article

The main purpose of this paper is to review the work on Runge-Kutta methods at the University of Toronto during the period 1963 to the present (1996). To provide some background, brief mention is also made of related work on the numerical solution of ordinary differential equations, but, with just a few exceptions, specific references are given only if the referenced material has a direct bearing on Runge-Kutta methods and their application to a variety of problem areas. There are several main themes. New Runge-Kutta formulas and new error control strategies are developed, leading for example to continuous methods and their application to areas such as delay, differential-algebraic and boundary-value problems. Software design and implementation are also emphasized. And so is the importance of careful testing and comparing. Other topics, such as the notion of effectiveness, taking advantage of parallelism, and handling discontinuities, are also discussed.