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

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