Strong Stability Preserving Multistep Runge-Kutta Methods

Abstract

High-order spatial discretizations with strong stability properties (such as monotonicity) are desirable for the solution of hyperbolic PDEs. Methods may be compared in terms of the strong stability preserving (SSP) time-step. We prove an upper bound on the SSP coefficient of explicit multistep Runge--Kutta methods of order two and above. Order conditions and monotonicity conditions for such methods are worked out in terms of the method coefficients. Numerical optimization is used to find optimized explicit methods of up to five steps, eight stages, and tenth order. These methods are tested on the advection and Buckley-Leverett equations, and the results for the observed total variation diminishing and positivity preserving time-step are presented.

Cited by in corpus (4)

• A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension
• A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
• Kershaw closures for linear transport equations in slab geometry II: high-order realizability-preserving discontinuous-Galerkin schemes
• A Strong Stability Preserving Analysis for Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions