High order positivity preserving and asymptotic preserving multi-derivative methods

In this work we present multi-derivative implicit-explicit (IMEX) Runge--Kutta schemes. We derive their order conditions up to third order, and show that such methods can preserve positivity (and more generally strong stability) with a time-step restriction independent of the stiff term, under mild assumptions on the operators. We present sufficient conditions under which such methods are positivity preserving and asymptotic preserving (AP) when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar-Gross-Krook (BGK) kinetic equation. Previous efforts to devise such methods have used an IMEX Runge--Kutta framework plus a second derivative final correction. In this work, we extend this approach to include derivative information at any stage of the computation. This multi-derivative IMEX approach allowed us to find a second order AP and positivity preserving method that improves upon previous work in terms of the allowable time-step size. Furthermore, this approach produces a third order method that is AP and positivity preserving for a time-step independent of the stiff term, a feature not possessed by any of the existing third-order IMEX schemes. We present numerical results to support the theoretical results, on a variety of problems.