A review of high order strong stability preserving two-derivative explicit, implicit, and IMEX methods

TL;DR

This review paper discusses the development of high-order strong stability preserving (SSP) multiderivative time-stepping methods, including explicit, implicit, and IMEX schemes, with conditions for stability and unconditionally SSP properties.

Contribution

It provides a comprehensive review of SSP theory for two-derivative Runge-Kutta and general linear methods, introducing novel second and third order methods with unconditional SSP properties.

Findings

Derived sufficient conditions for SSP in two-derivative methods

Presented SSP theory for explicit, implicit, and IMEX schemes

Introduced new second and third order methods with unconditional SSP

Abstract

High order strong stability preserving (SSP) time discretizations ensure the nonlinear non-inner-product strong stability properties of spatial discretizations suited for the stable simulation of hyperbolic PDEs. Over the past decade multiderivative time-stepping have been used for the time-evolution hyperbolic PDEs, so that the strong stability properties of these methods have become increasingly relevant. In this work we review sufficient conditions for a two-derivative multistage method to preserve the strong stability properties of spatial discretizations in a forward Euler and different conditions on the second derivative. In particular we present the SSP theory for explicit and implicit two-derivative Runge--Kutta schemes, and discuss a special condition on the second derivative under which these implicit methods may be unconditionally SSP. This condition is then used in the context of implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes, where the time-step restriction is independent of the stiff term. Finally, we present the SSP theory for implicit-explicit (IMEX) multi-derivative general linear methods, and some novel second and third order methods where the time-step restriction is independent of the stiff term.