A Strong Stability Preserving Analysis for Multistage Two-Derivative Time-Stepping Schemes Based on Taylor Series Conditions

TL;DR

This paper introduces SSP-TS methods that preserve strong stability in multistage two-derivative time-stepping schemes based on Taylor series, establishing their maximal order and demonstrating their effectiveness on PDEs.

Contribution

It develops sufficient conditions for SSP-TS methods, proves their maximal order is six, and provides optimized schemes with demonstrated advantages on hyperbolic PDEs.

Findings

SSP-TS methods can achieve order up to 6.

Optimized SSP-TS schemes outperform existing methods.

SSP-TS methods effectively preserve stability in PDE simulations.

Abstract

High order strong stability preserving (SSP) time discretizations are often needed to ensure the nonlinear (and sometimes non-inner-product) strong stability properties of spatial discretizations specially designed for the solution of hyperbolic PDEs. Multiderivative time-stepping methods have recently been increasingly used for evolving hyperbolic PDEs, and the strong stability properties of these methods are of interest. In our prior work we explored time discretizations that preserve the strong stability properties of spatial discretizations coupled with forward Euler and a second derivative formulation. However, many spatial discretizations do not satisfy strong stability properties when coupled with this second derivative formulation, but rather with a more natural Taylor series formulation. In this work we demonstrate sufficient conditions for an explicit two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and Taylor series formulation. We call these strong stability preserving Taylor series (SSP-TS) methods. We also prove that the maximal order of SSP-TS methods is p = 6, and define an optimization procedure that allows us to find such SSP methods. Several types of these methods are presented and their efficiency compared. Finally, these methods are tested on several PDEs to demonstrate the benefit of SSP-TS methods, the need for the SSP property, and the sharpness of the SSP time-step in many cases.