Numerical analysis of a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations with isotropic scattering

TL;DR

This paper develops and analyzes a spherical harmonic discontinuous Galerkin method for scaled radiative transfer equations, ensuring uniform convergence in diffusion regimes and providing optimal error estimates.

Contribution

It introduces a DG method for scaled radiative transfer equations with proven uniform convergence and optimal error bounds in diffusion limits.

Findings

The method converges uniformly in the small mean free path limit.

Error estimates of order (1 + O(ε))h^{k+1} are established.

Optimal convergence results are achieved on Cartesian grids.

Abstract

In highly diffusion regimes when the mean free path $\varepsilon$ tends to zero, the radiative transfer equation has an asymptotic behavior which is governed by a diffusion equation and the corresponding boundary condition. Generally, a numerical scheme for solving this problem has the truncation error containing an $\varepsilon^{-1}$ contribution, that leads to a nonuniform convergence for small $\varepsilon$. Such phenomenons require high resolutions of discretizations, which degrades the performance of the numerical scheme in the diffusion limit. In this paper, we first provide a--priori estimates for the scaled spherical harmonic ($P_N$) radiative transfer equation. Then we present an error analysis for the spherical harmonic discontinuous Galerkin (DG) method of the scaled radiative transfer equation showing that, under some mild assumptions, its solutions converge uniformly in $\varepsilon$ to the solution of the scaled radiative transfer equation. We further present an optimal convergence result for the DG method with the upwind flux on Cartesian grids. Error estimates of $\left(1+\mathcal{O}(\varepsilon)\right)h^{k+1}$ (where $h$ is the maximum element length) are obtained when tensor product polynomials of degree at most $k$ are used.