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 ffl 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 a ffl \Gamma1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small ffl, 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 ffl. 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. Key words. transfer equation, diffusion appro..