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.

翻译：在高扩散区域中，当平均自由程$\varepsilon$趋于零时，辐射传输方程具有由扩散方程及相应边界条件控制的渐近行为。通常，求解该问题的数值格式截断误差中包含$\varepsilon^{-1}$项，导致对小$\varepsilon$的非一致收敛。此类现象需要高分辨率的离散化，降低了数值格式在扩散极限下的性能。本文首先给出了缩放化球谐（$P_N$）辐射传输方程的先验估计，进而对缩放化辐射传输方程的球谐不连续伽辽金（DG）方法进行误差分析，证明在温和假设下其解在$\varepsilon$意义下一致收敛于缩放化辐射传输方程的解。我们进一步给出了笛卡尔网格上采用迎风通量的DG方法的最优收敛结果。当使用最高次数为$k$的张量积多项式时，获得了$\left(1+\mathcal{O}(\varepsilon)\right)h^{k+1}$（其中$h$为最大单元长度）的误差估计。