Abstract: Many kinetic transport models with collisions (including scattering) display multi-scale features due to the wide range of values of some parameters, such as the Knudsen number that is defined as the ratio of the mean free path and characteristic length of the problem. Depending on the parameter values, the problems can differ greatly in nature. Asymptotic preserving (AP) methods are a class of numerical schemes that are designed to work well with different scales or regimes of the models when the parameters vary, while capturing the correct asymptotic limits in the numerical level.

In this talk, we consider a kinetic transport equation under a diffusive scaling as a prototype model to study neutron transport or radiative transfer. As the Knudsen number goes to zero, our model approaches a diffusive equation. Our objective is to design high order AP methods，that are uniformly stable with respect to the Knudsen number, within the discontinuous Galerkin (DG) framework.  We will first examine and review the AP property of the upwind DG methods, then report on our developments to design AP DG methods following an entirely different route. The main ingredients of our methods include: reformulations of the model, high order DG methods in space, and suitably chosen high order implicit-explicit Runge-Kutta methods in time. In the presence of initial layers, a strategy is proposed to avoid accuracy reduction or loss. Computational complexity and theoretical findings will be discussed, along with some numerical examples.