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

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.