A block 5(4) explicit runge-kutta formula with "free" interpolation

Abstract

Embedded pairs of explicit Runge-Kutta formulae have been widely used for the numerical integration of non-stiff systems of first order ordinary differential equations. Because explicit Runge-Kutta formulae do not in general have a natural underlying interpolant, most implementations of these formulae restrict the steplength of integration so as to "hit" all output points exactly. Clearly this will normally lead to gross inefficiency when output is requested at many points and this is widely recognised as being a major disadvantage of explicit Runge-Kutta formulae. In addition there are some classes of problems for which an interpolation capability is indispensible. Recently the present author has proposed the use of block explicit Runge-Kutta formulae which advance the integration by more than one step at a time. One of the advantages of these block formulae is that they require less function evaluations per step than standard explicit Runge-Kutta formulae of the same order. In this paper we analyse completely the 5(4) two step Runge-Kutta formula with the minimum number of stages and show that it is possible to obtain a Runge-Kutta formula of this class with "free" interpolation capabilities. Some numerical results are given to compare the performance of a particular block 5(4) Runge-Kutta method with that of the widely used code RKF45 of Shampine and Watts.