Continuous extensions to high order runge-kutta methods

Abstract

Computer assisted derivation and improved techniques have led to effective explicit Runge-Kutta methods of higher order. These methods become inefficient when the step size must be reduced often to produce approximations at specified points. Considerable effort has been devoted to providing Runge-Kutta methods with an interpolation capability, so that approximations can be produced inexpensively at intermediate points of a successful step. New high order Hermite interpolants for two well known embedded Runge-Kutta methods of orders 7 and 8 are presented. These interpolants are constructed using values from two successive integration steps, are locally of O(h 8) or O(h 9), and require only one or four extra function evaluations per step respectively.