Before revision:  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.
After revision:  We present sufficient conditionsunder which such 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 . Previous efforts to devise such methods have used an IMEX Runge--Kutta framework plus a second derivative final correction.