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

TL;DR

This paper introduces high-order, unconditionally SSP implicit and IMEX Runge--Kutta methods with asymptotic preserving properties, enabling stable and efficient time integration for stiff problems without timestep restrictions.

Contribution

The paper develops the first unconditionally SSP implicit and IMEX Runge--Kutta schemes of order up to 4 and 3 respectively, with new order conditions and asymptotic preserving features.

Findings

Unconditionally SSP schemes of order up to 4 developed.

IMEX schemes of order up to 3 constructed.

Numerical tests confirm stability and asymptotic properties.

Abstract

In this work we present a class of high order unconditionally strong stability preserving (SSP) implicit multi-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. We present numerical results to support the theoretical results, on a variety of problems.