Robust defect control with Runge-Kutta schemes

Enright [Numerical Analysis Report 122, University of Manchester, Manchester, U.K., 1986] implements a Runge–Kutta method for solving the initial value problem using an alternative to the standard local error control scheme. The aim is to control the defect associated with a local interpolant by sampling its value at one or more fixed points within each step. However, in general, the quality of a sample point is problem-dependent and also varies from step to step. Two classes of interpolant are presented for which the asymptotic behaviour of the defect is known a priori, allowing optimal sample points to be chosen.