  Discontinuous Galerkin (DG) methods are widely adopted to discretize the
radiation transport equation (RTE) with diffusive scalings. One of the most
important advantages of the DG methods for RTE is their asymptotic preserving
(AP) property, in the sense that they preserve the diffusive limits of the
equation in the discrete setting, without requiring excessive refinement of the
discretization. However, compared to finite element methods or finite volume
methods, the employment of DG methods also increases the number of unknowns,
which requires more memory and computational time to solve the problems. In
this paper, when the spherical harmonic method is applied for the angular
discretization, we perform an asymptotic analysis which shows that to retain
the uniform convergence, it is only necessary to employ non-constant elements
for the degree zero moment only in the DG spatial discretization. Based on this
observation, we propose a heterogeneous DG method that employs polynomial
spaces of different degrees for the degree zero and remaining moments
respectively. To improve the convergence order, we further develop a spherical
harmonics hybrid DG finite volume method, which preserves the AP property and
convergence rate while tremendously reducing the number of unknowns. Numerical
examples are provided to illustrate the effectiveness and accuracy of the
proposed scheme.
