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 conditionsunder 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.
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.
A
a class of high order unconditionally strong stability preserving (SSP) implicit
A
Runge--Kutta schemes, and SSP
A
multi-derivative
R
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
schemes where the
A
is
R
, under mild assumptions on the operators. We present sufficient conditionsunder which such
. 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
R
and asymptotic preserving (AP)
, and we present sufficient conditions under which such methods are also asymptotic preserving
D
. 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