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.