Pseudo-Runge-Kutta Methods of the Fifth Order

A family of fifth-order pseudo-Runge-Kutta methods for the numerical solution of systems of ordinary differential equations is presented. A procedure for determining an “optimal” set of parameters is given, and several examples are considered. The principal advantage of these methods is that, for a fixed stepsize, they require two less function evaluations at each step than do the corresponding fifth-order Runge-Kutta methods. Their principal disadvantage is that they are not self-starting; they require two initial values. Numerically, pseudo-Runge-Kutta and Runge-Kutta methods seem to be comparable.