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

TL;DR

This paper analyzes the uniform convergence of an upwind discontinuous Galerkin method for scaled radiative transfer equations, showing convergence properties and strategies to handle boundary layers in slab geometries.

Contribution

It provides the first error analysis demonstrating uniform convergence of DG methods for scaled radiative transfer equations with insights into boundary layer effects.

Findings

DG method converges uniformly with respect to the scattering parameter

Optimal convergence achieved in 1D without boundary layers

Numerical tests confirm theoretical results

Abstract

We present an error analysis for the discontinuous Galerkin method applied to the discrete-ordinate discretization of the steady-state radiative transfer equation. Under some mild assumptions, we show that the DG method converges uniformly with respect to a scaling parameter $\varepsilon$ which characterizes the strength of scattering in the system. However, the rate is not optimal and can be polluted by the presence of boundary layers. In one-dimensional slab geometries, we demonstrate optimal convergence when boundary layers are not present and analyze a simple strategy for balance interior and boundary layer errors. Some numerical tests are also provided in this reduced setting.