  In this paper, we develop high-order asymptotic preserving (AP) schemes for
the BGK equation in a hyperbolic scaling, which leads to the macroscopic models
such as the Euler and compressible Navier-Stokes equations in the asymptotic
limit. Our approaches are based on the so-called micro-macro formulation of the
kinetic equation which involves a natural decomposition of the problem to the
equilibrium and the non-equilibrium parts. The proposed methods are formulated
for the BGK equation with constant or spatially variant Knudsen number. The new
ingredients for the proposed methods to achieve high order accuracy are the
following: we introduce discontinuous Galerkin (DG) discretization of arbitrary
order of accuracy with nodal Lagrangian basis functions in space; we employ a
high order globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta (RK)
scheme as time discretization. Two versions of the schemes are proposed: Scheme
I is a direct formulation based on the micro-macro decomposition of the BGK
equation, while Scheme II, motivated by the asymptotic analysis for the
continuous problem, utilizes certain properties of the projection operator.
Compared with Scheme I, Scheme II not only has better computational efficiency
(the computational cost is reduced by half roughly), but also allows the
establishment of a formal asymptotic analysis. Specifically, it is demonstrated
that when $0<\epsilon \ll 1$, Scheme II, up to $\mathcal{O}(\epsilon^2)$,
becomes a local DG discretization with an explicit RK method for the
macroscopic compressible Navier-Stokes equations, a method in a similar spirit
to the ones in [Bassi \& Rabey 1997, Cockburn \& Shu 1998]. Numerical results
are presented for a wide range of Knudsen number to illustrate the
effectiveness and high order accuracy of the methods.
