Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

TL;DR

This paper develops explicit multistage two-derivative SSP time-stepping schemes, including a novel fifth order method, improving stability and accuracy for hyperbolic PDEs compared to traditional methods.

Contribution

It introduces sufficient conditions for SSP multistage two-derivative methods and constructs the first explicit five-order SSP method with three stages.

Findings

Designed a three-stage fifth order SSP method.

Verified order of convergence on scalar PDEs.

Highlighted the importance of SSP conditions and time-step sharpness.

Abstract

High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability time-stepping methods with large allowable strong stability time-step has been an active area of research over the last two decades. Recently, multiderivative time-stepping methods have been implemented with hyperbolic PDEs. In this work we describe sufficient conditions for a two-derivative multistage method to be SSP, and find some optimal SSP multistage two-derivative methods. While explicit SSP Runge--Kutta methods exist only up to fourth order, we show that this order barrier is broken for explicit multi-stage two-derivative methods by designing a three stage fifth order SSP method. These methods are tested on simple scalar PDEs to verify the order of convergence, and demonstrate the need for the SSP condition and the sharpness of the SSP time-step in many cases.