  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.
