A comparative analysis of explicit, IMEX and implicit strong stability preserving Runge-Kutta schemes

Abstract

Selecting between explicit and implicit time marching schemes, as well as non-iterative explicit-implicit ones, is well understood on a linear stability basis. Implicit schemes are favored when either searching for steady-states or performing time accurate simulations of stiff equations due to their larger time steps. When nonlinear stability properties must be preserved, however, solution monotonicity is guaranteed by a time step restriction that puts into question the usefulness of implicit schemes. Five test cases are used in this paper for a comparative efficiency analysis of explicit, implicit and implicit-explicit Strong Stability Preserving (SSP) Runge-Kutta (RK) schemes. They indicate that explicit SSP-RK schemes are, in general, the most efficient ones. It is important to emphasize that great care must be exercised when generalizing these conclusions, since such an analysis is inherently software, hardware, numerics and model dependent. Nevertheless, this study provides the first systematic evidence in the literature confirming these trends, to the best of our knowledge. Finally, implicit-explicit SSP-RK schemes do perform well, even though they are not optimal. Hence, there is a need for the development of their optimal versions before any conclusions can be drawn about their efficiency.