Efficient SSP low-storage Runge–Kutta methods

Abstract

In this paper we study the efficiency of Strong Stability Preserving (SSP) Runge–Kutta methods that can be implemented with a low number of registers using their Shu–Osher representation. SSP methods have been studied in the literature and stepsize restrictions that ensure numerical monotonicity have been found. However, for some problems, the observed stepsize restrictions are larger than the theoretical ones. Aiming at obtaining additional properties of the schemes that may explain their efficiency, in this paper we study the influence of the local error term in the observed stepsize restrictions. For this purpose, we consider the family of 5-stage third order SSP explicit Runge–Kutta methods, namely SSP(5,3), and the Buckley–Leverett equation. We deal with optimal SSP(5,3) schemes whose implementation requires at least 3 memory registers, and non-optimal 2-register SSP(5,3) schemes. The numerical experiments done show that small error constants improve the efficiency of the method in the sense that larger observed SSP coefficients are obtained.