  The $M_1$ minimum entropy moment system is a system of hyperbolic balance
laws that approximates the radiation transport equation, and has many desirable
properties. Among them are symmetric hyperbolicity, entropy decay, moment
realizability, and correct behavior in the diffusion and free-streaming limits.
However, numerical difficulties arise when approximating the solution of the
$M_1$ model by high order numerical schemes; namely maintaining the
realizability of the numerical solution and controlling spurious oscillations.
In this paper, we extend a previously constructed one-dimensional realizability
limiting strategy to 2D. In addition, we perform a numerical study of various
combinations of the realizability limiter and the TVBM local slope limiter on a
third order Discontinuous Galerkin (DG) scheme on both triangular and
rectangular meshes. In several test cases, we demonstrate that in general, a
combination of the realizability limiter and a TVBM limiter is necessary to
obtain a robust and accurate numerical scheme. Our code is published so that
all results can be reproduced by the reader.
