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

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.

Authors: Raymond J. Spiteri, Steven J. Ruuth

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