On the economization of explicit Runge-Kutta methods

A modification to explicit Runge-Kutta (RK) methods is proposed. Schemes are constructed which require less derivative-evaluations to achieve a certain order than the classical RK methods do. As an example, we give a second-order method requiring one evaluation, two third-order methods using one and two evaluations, respectively and finally a fourth-order method which requires two evaluations. Numerical examples illustrate the behaviour of these schemes.