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Diffusive scaled linear kinetic equations appear in various applications, and they contain a small parameter ɛ 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 ɛ, with the time step restriction being independent of ɛ. 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 ɛ, for all the models. The qualitative results for diffusion asymptotics, and equilibrium and non-equilibrium inflow boundaries are also presented. [ABSTRACT FROM AUTHOR]

We develop generalized polynomial chaos (gPC) based stochastic Galerkin (SG) methods for a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focus on a highly oscillatory scalar model with random uncertainty. Our method is built upon the nonlinear geometrical optics (NGO) based method, developed in Crouseilles et al. [Math. Models Methods Appl. Sci.23 (2017) 2031–2070] for numerical approximations of deterministic equations, which can obtain accurate pointwise solution even without numerically resolving spatially and temporally the oscillations. With the random uncertainty, we show that such a method has oscillatory higher order derivatives in the random space, thus requires a frequency dependent discretization in the random space. We modify this method by introducing a new "time" variable based on the phase, which is shown to be non-oscillatory in the random space, based on which we develop a gPC-SG method that can capture oscillations with the frequency-independent time step, mesh size as well as the degree of polynomial chaos. A similar approach is then extended to a semiclassical surface hopping model system with a similar numerical conclusion. Various numerical examples attest that these methods indeed capture accurately the solution statistics pointwisely even though none of the numerical parameters resolve the high frequencies of the solution. [ABSTRACT FROM AUTHOR]