Estimating local truncation errors for Runge-Kutta methods

Abstract

As an alternative to the use of embedded formulas, it is proposed that local truncation errors might be estimated by a generalization of the method of Ceschino and Kuntzmann (1963). Computed solution values over several successive steps, together with computed derivatives, are used to obtain an accurate approximation to the local truncation error using a Hermite interpolation formula. In this paper, it is shown how a variable-stepsize adaptation of this approximation can be generated cheaply as the solution proceeds. Because the Hermite interpolant will always be available when this procedure is in use, dense output is also available at little additional cost.