A Numerical Study of Diagonally Split Runge-Kutta Methods for PDEs with Discontinuities

Diagonally split Runge–Kutta (DSRK) time discretization methods are a class of implicit time-stepping schemes which offer both high-order convergence and a form of nonlinear stability known as unconditional contractivity. This combination is not possible within the classes of Runge–Kutta or linear multistep methods and therefore appears promising for the strong stability preserving (SSP) timestepping community which is generally concerned with computing oscillation-free numerical solutions of PDEs. Using a variety of numerical test problems, we show that although secondand third-order unconditionally contractive DSRKmethods do preserve the strong stability property for all time step-sizes, they suffer from order reduction at large step-sizes. Indeed, for time-steps larger than those typically chosen for explicit methods, these DSRK methods behave like firstorder implicit methods. This is unfortunate, because it is precisely to Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, V5A1S6 Canada ([email protected]). The work of this author was partially supported by an NSERC Canada PGS-D scholarship, a grant from NSERC Canada, and a scholarship from the Pacific Institute for the Mathematical Sciences (PIMS). Department of Mathematics, University of Massachusetts Dartmouth, North Dartmouth MA 02747 USA ([email protected]). This work was supported by AFOSR grant number FA9550-06-1-0255. 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.