Midway reassignment of Runge–Kutta step length

Abstract

At the end of each Runge–Kutta (RK) step, a new step‐size is predicted, and later, during the evaluations of the next step, if we feel that the prediction was not correct, nothing can be done. Here, we suggest an intermediate error estimation of low order, at the beginning of the integration, that can be used for the reconsideration of the step length before this is completed. This requires the evaluation of a variable coefficient RK. These coefficients depend on the value , where is the new prediction of the step‐size. In addition, a new control for the step length at the end of each integration is designed taking advantage from the extra estimator. The basic concept of this idea is to produce general purpose RK methods that have the capability to reduce the number of step rejections for difficult problems (such as orbits with high eccentricities and Van der Pole equation).