  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.
