High order asymptotic preserving scheme for diffusive scaled linear kinetic equations with general initial conditions

Diffusive scaled linear kinetic equations appear in various applications, and they contain a small parameter $\epsilon$ that forces a severe time step restriction for standard explicit schemes. Asymptotic preserving (AP) schemes are those schemes that attain asymptotic consistency and uniform stability for all values of ε, with the time step restriction being independent of ε. In this work, we develop high order AP scheme for such diffusive scaled kinetic equations with both well-prepared and non-well-prepared initial conditions by employing IMEX-RK time integrators such as CK-ARS and A types. This framework is also extended to a different collision model involving advection-diffusion asymptotics, and the AP property is proved formally. A further extension of our framework to inflow boundaries has been made, and the AP property is verified. The temporal and spatial orders of accuracy of our framework are numerically validated in different regimes of ε, for all the models. The qualitative results for diffusion asymptotics, and equilibrium and non-equilibrium inflow boundaries are also presented.