Economical handling of Runge–Kutta–Nyström step rejection

Abstract

Runge–Kutta–Nyström (RKN) pairs are amongst the most popular methods for solving numerically Initial Value Problems of second order. Here we consider the special case with first derivative absent from the right hand side of the classical definition of this problem. While using a RKN pair, we may experience rejection of some steps through the interval of integration. Traditionally, the whole amount of evaluations are then dropped and we proceed with a completely new round of computations. In this work we propose avoiding this waste and continue by using again the rejected RKN stages. We especially focus on a RKN pair of orders six and four. After a step rejection we use again all the previously evaluated stages and add only one new stage. We proceed by evaluating the output in a smaller step. By this technique we manage to reduce significantly the cost in a set of problems that are known to pose difficulties in RKN step size changing algorithms.