Block Runge-Kutta methods for the numerical integration of initial value problems in ordinary differential equations. I. The nonstiff case

Abstract

Block Runge-Kutta formulae suitable for the approximate numerical integration of initial value problems for first order systems of ordinary differential equations are derived. Considered in detail are the problems of varying both order and stepsize automatically. This leads to a class of variable order block explicit Runge-Kutta formulae for the integration of nonstiff problems and a class of variable order block implicit formulae suitable for stiff problems. The central idea is similar to one due to C. W. Gear in developing Runge-Kutta starters for linear multistep methods. Some numerical results are given to illustrate the algorithms developed for both the stiff and nonstiff cases and comparisons with standard Runge-Kutta methods are made.