Fifth-order Runge-Kutta with higher order derivative approximations.

Given $y'=f(y)$, standard Runge-Kutta methods perform multiple evaluations of $f(y)$ in each integration sub-interval as required for a given accuracy. Evaluations of $y''=f_{y}f$ or higher derivatives are not considered due to the assumption that the calculations involved in these functions exceed those of $f$. However, $y''$ can be approximated to sufficient accuracy from past and current evaluations of $f$ to achieve a higher order of accuracy than is available through current functional evaluations alone. In July of 1998 at the ANODE (Auckland Numerical Ordinary Differential Equations) Workshop, we introduced a new class of Runge-Kutta methods based on this observation (Goeken~1999). We presented a third-order method which requires only two evaluations of $f$ and a fourth-order method which requires three. This paper reviews these two methods and gives the general solution to the equations generated by the fifth-order methods of this new class. Interestingly, these fifth-order methods require only four functional evaluations per step whereas standard Runge-Kutta methods require six.