New Asymptotic Preserving, Hybrid Discontinuous Galerkin Methods the Radiation Transport Equation with Isotropic Scattering and Diffusive Scaling

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.