Error propagation in Runge-Kutta methods

Abstract

This paper deals with the general (explicit or implicit) Runge-Kutta method for the numerical solution of initial value problems. We consider how perturbations (like rounding errors), introduced in the consecutive time steps of the method, are propagated during the calculations. The paper reviews various classical estimates, obtained since Runge (1905), of the accumulated error generated by all perturbations together. These estimates are compared with each other, regarding their behaviour as the step size h tends to zero. Two new theorems are presented which extend and improve some of the classical results. The first of these theorems gives an upperbound for the norm of the accumulated error, whereas the second shows that this upperbound is maximally refined with respect to its behaviour as h tends to zero. The paper also reviews various error estimates, obtained since the seventies, which are still relevant in some notorious (stiff) cases where h does not necessarily tend to zero and where the estimates mentioned above significantly overestimate the actual error. The focus in the paper is on estimating the accumulated error with an arbitrary norm in R s (not necessarily generated by an inner product).