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A class of high order asymptotic preserving (AP) schemes has been developedfor the BGK equation in Xiong et. al. (2015) [37], which is based on themicro-macro formulation of the equation. The nodal discontinuous Galerkin (NDG)method with Lagrangian basis functions for spatial discretization and globallystiffly accurate implicit-explicit (IMEX) Runge-Kutta (RK) scheme as timediscretization are introduced with asymptotic preserving properties. However,it is only necessary to solve the kinetic equation when the hydrodynamicdescription breaks down. Motivated by the recent work in Filbet and Rey (2015)[23], it is more naturally to construct a hierarchy scheme under the NDG-IMEXframework without hybridization, as the formal analysis in [37] shows that when$\epsilon$ is small, the NDG-IMEX scheme becomes a local discontinuous Galerkin(LDG) scheme for the compressible Navier-Stokes equations, and when$\epsilon=0$ it is a discontinuous Galerkin (DG) scheme for the compressibleEuler equations. Moveover, we propose to combine the kinetic regime with thehydrodynamic regime including both the compressible Euler and Navier-Stokesequations. Numerical experiments demonstrate very decent performance of the newapproach. In our numerics, all three regimes are clearly divided, leading togreat savings in terms of the computational cost.

High Order Hierarchical Asymptotic Preserving Nodal Discontinuous Galerkin IMEX Schemes For The BGK Equation