The Convergence of Numerical Transfer Schemes in Diffusive Regimes I: Discrete-Ordinate Method

In highly diffusive regimes, the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path $\epsilon$ tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis. A numerical scheme for solving this problem has an $\epsilon^{-1}$ contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small $\epsilon$, thus degrading its performance in diffusive regimes. In this paper we show that whenever the discrete-ordinate method has the correct diffusion limit, both in the interior and at the boundaries, its solutions converge to the solution of the transport equation uniformly in $\epsilon$. Our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete-ordinate method.