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In this work we present a class of high order unconditionally strongstability preserving (SSP) implicit multi-derivative Runge--Kutta schemes, andSSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where thetime-step restriction is independent of the stiff term. The unconditional SSPproperty for a method of order $p>2$ is unique among SSP methods, and dependson a backward-in-time assumption on the derivative of the operator. We showthat this backward derivative condition is satisfied in many relevant caseswhere SSP IMEX schemes are desired. We devise unconditionally SSP implicitRunge--Kutta schemes of order up to $p=4$, and IMEX Runge--Kutta schemes oforder up to $p=3$. For the multi-derivative IMEX schemes, we also derive andpresent the order conditions, which have not appeared previously. Theunconditional SSP condition ensures that these methods are positivitypreserving, and we present sufficient conditions under which such methods arealso asymptotic preserving when applied to a range of problems, including ahyperbolic relaxation system, the Broadwell model, and theBhatnagar-Gross-Krook (BGK) kinetic equation. We present numerical results tosupport the theoretical results, on a variety of problems.

High order strong stability preserving multi-derivative implicit and IMEX Runge--Kutta methods with asymptotic preserving properties