Construction of the ef-based Runge–Kutta methods revisited

The purpose of this paper is to revisit the exponential fitting (ef) technique when building up Runge–Kutta methods for solving ordinary differential equations. We propose a modification in such a way that the contamination of the final stage by the errors produced in the internal stages becomes visible. The modified technique is illustrated on a simple version, namely the two-stage explicit Runge–Kutta method, for which we obtain new expressions for the coefficients. The version obtained in this way is then compared for accuracy and stability with that obtained by means of the standard ef technique.