Strong stability preserving two-step Runge-Kutta methods

TL;DR

This paper studies strong stability preserving two-step Runge-Kutta methods, proving their properties, deriving order conditions, and presenting high-order methods with improved SSP coefficients demonstrated through numerical examples.

Contribution

It establishes the subclass of SSP TSRK methods without previous step stages, derives order conditions, and introduces high-order methods with larger SSP coefficients.

Findings

Explicit SSP TSRK methods can achieve up to eighth order.

New methods have larger SSP coefficients than existing methods of same order.

Numerical examples demonstrate effectiveness in high-order WENO discretizations.

Abstract

We investigate the strong stability preserving (SSP) property of two-step Runge-Kutta (TSRK) methods. We prove that all SSP TSRK methods belong to a particularly simple subclass of TSRK methods, in which stages from the previous step are not used. We derive simple order conditions for this subclass. Whereas explicit SSP Runge-Kutta methods have order at most four, we prove that explicit SSP TSRK methods have order at most eight. We present TSRK methods of up to eighth order that were found by numerical search. These methods have larger SSP coefficients than any known methods of the same order of accuracy, and may be implemented in a form with relatively modest storage requirements. The usefulness of the TSRK methods is demonstrated through numerical examples, including integration of very high order WENO discretizations.