The Relative Efficiency of Quenching and Reintegration for Global Error Control in Runge-Kutta Methods

Abstract

We compare the efficiency of two algorithms for controlling global error in Runge-Kutta methods, i.e. reintegration and RKQ. We find that RKQ (which uses a quenching strategy) is, on average, more efficient than reintegration, by at least 16% or 30%, depending on the form of RKQ. Some numerical examples indicate that larger gains in efficiency are possible. The Runge-Kutta methods we consider are explicit, implicit (Radau, Lobatto, Gauss), embedded and Nyström.