An improved class of generalized Runge–Kutta–Nyström methods for special second-order differential equations

Abstract

We propose a new family of explicit methods of order four with two evaluations per step, for the numerical integration of special second-order differential equations given by y ″= f( y). These two-stage formulas can be seen as a generalization of the explicit two-stage Runge–Kutta–Nyström methods, providing better order and stability results. We will show that it is possible to obtain methods that are more efficient than the classical Runge–Kutta–Nyström one-step methods with the same number of evaluations per step, specially when highly oscillatory problems are considered. Some numerical experiments are discussed in order to show the good performance of the new schemes.