Runge-Kutta with higher order derivative approximations

Abstract We introduce a form of Runge–Kutta in which it is assumed that the user will evaluate both f and f′ in solving y′=f(x,y) numerically. This allows us to introduce new Runge–Kutta parameters that increase the order of accuracy of the solution with evaluations of f′ replacing evaluations of f. If f′ is approximated to sufficient accuracy from past and current evaluations of f, rather than calculated exactly, the order of convergence is retained. The resulting multi-step Runge–Kutta method can be thought of as replacing functional evaluations with approximations of f′. Normally, this is an attractive option since f′ can be approximated to the required accuracy with little arithmetic. Here we present an O(h3) method which requires only two evaluations of f and an O(h4) which requires three.