Invited Lecture
High order strong stability preserving multi-derivative implicit and IMEX Runge-Kutta methods
University of Massachusetts Dartmouth, USA
Abstract
In this talk we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge-Kutta schemes, and SSP implicit-explicit (IMEX) multi-derivative Runge-Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods, and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge-Kutta schemes of order up to $p=4$, and IMEX Runge-Kutta schemes of order up to $p=3$. For the multi-derivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation.
This is joint work with Zachary Grant, Jingwei Hu, and Ruiwen Shu