The development of Runge-Kutta methods for partial differential equations

Abstract

A widely-used approach in the time integration of initial-value problems for time-dependent partial differential equations (PDEs) is the method of lines. This method transforms the PDE into a system of ordinary differential equations (ODEs) by discretization of the space variables and uses an ODE solver for the time integration. Since ODEs originating from space-discretized PDEs have a special structure, not every ODE solver is appropriate. For example, the well-known fourth-order Runge-Kutta method is highly inefficient if the PDE is parabolic, but it performs often quite satisfactory if the PDE is hyperbolic. In this lecture, we give a survey of the development of ODE methods that are tuned to space-discretized PDEs. Because of the overwhelming number of methods that have been proposed through the years, we confine our considerations to Runge-Kutta type methods. In this contribution to the historical surveys presented at the IMACS 14th World Congress held in July 1994 in Atlanta, we describe work of Crank and Nicolson (1947), Laasonen (1949), Peaceman and Rachford (1955), Yuan' Chzao-Din (1958), Stiefel (1958), Franklin (1959), Guillou and Lago (1960), Metzger (1967), Lomax (1968), Gourlay (1970), Riha (1972), Gentzsch and Schlüter (1978), Vichnevetsky (1983), Kinnmark and Gray (1984), Sonneveld and van Leer (1985), as well as research carried out at CWI.