  In this paper, we will develop a class of high order asymptotic preserving
(AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray
radiative transfer equations (GRTEs). Inspired by the work
\cite{Peng2020stability}, in which stability enhanced high order AP DG methods
are proposed for linear transport equations, we propose to pernalize the
nonlinear GRTEs under the micro-macro decomposition framework by adding a
weighted linear diffusive term. In the diffusive limit, a hyperbolic, namely
$\Delta t=\mathcal{O}(h)$ where $\Delta t$ and $h$ are the time step and mesh
size respectively, instead of parabolic $\Delta t=\mathcal{O}(h^2)$ time step
restriction is obtained, which is also free from the photon mean free path. The
main new ingredient is that we further employ a Picard iteration with a
predictor-corrector procedure, to decouple the resulting global nonlinear
system to a linear system with local nonlinear algebraic equations from an
outer iterative loop. Our scheme is shown to be asymptotic preserving and
asymptotically accurate. Numerical tests for one and two spatial dimensional
problems are performed to demonstrate that our scheme is of high order,
effective and efficient.
