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.

翻译：高阶强稳定保持（SSP）时间离散化方法能够确保空间离散化在非线性非内积意义下的强稳定性，这类空间离散化适用于双曲型偏微分方程的稳定数值模拟。过去十年中，多导数时间步进方法已广泛应用于双曲型偏微分方程的时间演化计算，使得这类方法的强稳定性研究日益重要。本文系统综述了多级两步导数方法保持空间离散化强稳定性的充分条件，这些条件分别基于前向欧拉格式与二阶导数的不同约束。特别地，我们建立了显式与隐式两步导数Runge-Kutta格式的SSP理论，并讨论了使隐式方法具备无条件SSP特性的二阶导数特殊约束条件。进一步地，我们将该条件应用于隐显式（IMEX）多导数Runge-Kutta格式，使得时间步长限制与刚性项无关。最后，我们提出了隐显式（IMEX）多导数广义线性方法的SSP理论，并构造了若干时间步长限制独立于刚性项的二阶与三阶新方法。