  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.
