ON THE LACK OF CONVERGENCE OF UNCONDITIONALLY STABLE EXPLICIT RATIONAL RUNGE-KUTTA SCHEMES

The poor performance of the rational Runge-Kutta (RRK) schemes of E. Hairer are investigated. By considering two simple model problems, it is demonstrated that this poor performance is in fact due to a lack of convergence. A conceptual model of an unconditionally stable implicit-explicit time-integration scheme is also considered. With the aid of this model, it is possible to establish necessary bounds on the extent of the explicit region for convergence. This demonstrates the limited applicability of such hybrid time-integration schemes.