  A class of high order asymptotic preserving (AP) schemes has been developed
for the BGK equation in Xiong et. al. (2015) [37], which is based on the
micro-macro formulation of the equation. The nodal discontinuous Galerkin (NDG)
method with Lagrangian basis functions for spatial discretization and globally
stiffly accurate implicit-explicit (IMEX) Runge-Kutta (RK) scheme as time
discretization are introduced with asymptotic preserving properties. However,
it is only necessary to solve the kinetic equation when the hydrodynamic
description 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-IMEX
framework 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 compressible
Euler equations. Moveover, we propose to combine the kinetic regime with the
hydrodynamic regime including both the compressible Euler and Navier-Stokes
equations. Numerical experiments demonstrate very decent performance of the new
approach. In our numerics, all three regimes are clearly divided, leading to
great savings in terms of the computational cost.
