A Spherical Harmonic Discontinuous Galerkin Method for Radiative Transfer Equations with Vacuum Boundary Conditions

Uniform convergence of an upwind discontinuous Galerkin method for solving scaled discrete-ordinate radiative transfer equations with isotropic scattering

“…The treatment of such boundary conditions is possible with direct approaches [8] and variational frameworks [15,34]. For vacuum (zero inflow) conditions, an extended computational domain can be employed to maintain sparsity of the P N system [46,16] or to handle complicated geometries [40] in a periodic setting.…”

“…Spatial discretizations strategies for the P N equations include finite differences [43], least squares [7,47] and mixed finite elements [16], discontinuous Galerkin methods [46], and psuedo-spectral methods [40]. Methods for capturing the diffusion limit for the time-dependent P N equations [25] have been explored in both finite-volume [24] and discontinuous Galerkin (DG) [37,36] contexts; and a rigorous analysis of the steady-state problem, based on a least-squares finite element formulation, can be found in [35].…”

“…DG methods were first introduced in [41] to simulate transport equations like (1), and a rigorous analysis was carried out in [31] for a simplified equation without scattering. A complete space-angle convergence analysis of discrete ordinate DG methods was conducted in [22] for problems with scattering, and analogous results for the spherical harmonic DG method were recently proved in [46]. While the analysis studied in [22,46] is valid for any fixed ε = O(1), it does not address the behavior of the DG method when ε → 0.…”

“…A complete space-angle convergence analysis of discrete ordinate DG methods was conducted in [22] for problems with scattering, and analogous results for the spherical harmonic DG method were recently proved in [46]. While the analysis studied in [22,46] is valid for any fixed ε = O(1), it does not address the behavior of the DG method when ε → 0. In the context of discrete ordinates angular discretization, it is known that DG methods can capture the diffusion limit whenever the underlying approximation space supports globally continuous linear polynomials [29,1,20].…”

“…In this paper, we investigate the convergence of P N -DG methods for the scaled RTE with isotropic scattering and periodic boundary conditions. The analysis builds upon ideas from [46] for establishing angular error estimates and from [45] for establishing spatial error estimates. The main contribution is to derive spectral estimates for the P N discretization that are uniform in ε as well as O(h k ) error estimates for the DG scheme that are uniform in ε whenever the DG space contains Q 1 for Cartesian cells and P 1 for triangles.…”