  We implement a high-order numerical scheme for the entropy-based moment
closure, the so-called M$_N$ model, for linear kinetic equations in slab
geometry. A discontinuous Galerkin (DG) scheme in space along with a
strong-stability preserving Runge-Kutta time integrator is a natural choice to
achieve a third-order scheme, but so far, the challenge for such a scheme in
this context is the implementation of a linear scaling limiter when the
numerical solution leaves the set of realizable moments (that is, those moments
associated with a positive underlying distribution). The difficulty for such a
limiter lies in the computation of the intersection of a ray with the set of
realizable moments. We avoid this computation by using quadrature to generate a
convex polytope which approximates this set. The half-space representation of
this polytope is used to compute an approximation of the required intersection
straightforwardly, and with this limiter in hand, the rest of the DG scheme is
constructed using standard techniques. We consider the resulting numerical
scheme on a new manufactured solution and standard benchmark problems for both
traditional M$_N$ models and the so-called mixed-moment models. The
manufactured solution allows us to observe the expected convergence rates and
explore the effects of the regularization in the optimization.
