Non-linear evolution using optimal fourth-order strong-stability-preserving Runge-Kutta methods

Authors:
Raymond J. Spiteri;Steven J. Ruuth
Affiliations:
Department of Computer Science, Dalhousie University, Halifax, NS, Canada;Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
Venue:
Mathematics and Computers in Simulation - Nonlinear waves: computation and theory II
Year:
2003

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Abstract

Strong-stability-preserving (SSP) time discretization methods (also known as total-variation-diminishing or TVD methods) are popular and effective algorithms for the simulation of partial differential equations having discontinuous or shock-like solutions. Optimal SSP Runge-Kutta (SSPRK) schemes have been previously found for methods with up to five stages and up to fourth order. In this paper, we present new optimal fourth-order SSPRK schemes with mild storage requirements and up to eight stages. We find that these schemes are ultimately more efficient than the known fourth-order SSPRK schemes because the increase in the allowable time-step more than offsets the added computational expense per step. We demonstrate these efficiencies on a pair of scalar conservation laws.