Explicit Runge-Kutta Methods for Stiff Problems with a Gap in Their Eigenvalue Spectrum

In this paper we consider the numerical solution of stiff problems in which the eigenvalues are separated into two clusters, one containing the "stiff", or fast, components and one containing the slow components, that is, there is a gap in their eigenvalue spectrum. By using exponential fitting techniques we develop a class of explicit Runge-Kutta methods, that we call stability fitted methods, for which the stability domain has two regions, one close to the origin and the other one fitting the large eigenvalues. We obtain the size of their stability regions as a function of the order and the fitting conditions. We also obtain conditions that the coefficients of these methods must satisfy to have a given stiff order for the Prothero-Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments.