Optimal Implicit Strong Stability Preserving Runge–Kutta Methods

Strong stability preserving (SSP) time discretizations were developed for use with the spatial discretization of partial differential equations that are strongly stable under forward Euler time integration. SSP methods preserve convex boundedness and contractivity properties satisfied by forward Euler, under a modified time-step restriction. We turn to implicit Runge–Kutta methods to alleviate this time step restriction, and present implicit strong stability preserving (SSP) Runge–Kutta methods which are optimal in the sense that they preserve convex boundedness properties under the largest timestep possible among all methods with a given number of stages and order of accuracy. We consider methods up to order six (the maximal order of implicit SSP methods) and up to eleven stages. The numerically optimal Runge–Kutta methods found are all diagonally implicit, leading us to conjecture that optimal implicit SSP Runge–Kutta methods are diagonally implicit. These methods allow a significant increase in SSP time-step limit, compared to explicit methods of the same order and Department of Applied Mathematics, University of Washington, Seattle, WA 981952420 ([email protected]). The work of this author was funded by a U.S. Dept. of Energy Computational Science Graduate Fellowship. Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A1S6 Canada ([email protected]). The work of this author was partially supported by a grant from NSERC Canada and a scholarship from the Pacific Institute of Mathematics (PIMS). Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth MA 02747. This work was supported by AFOSR grant number FA9550-06-1-0255