The asymptotic diffusion limit of continuous and discrete steady-state multigroup radiative transfer problems

The authors consider a linear steady-state multigroup, radiative transfer problem in which the opacities are large [O(ε-1)] and interior sources are small [ O(ε)]. An asymptotic analysis of this problem as ε → 0 leads to a one-group diffusion equation (related to the equilibrium diffusion equation) in the interior of the system, plus boundary conditions for this equation. The same asymptotic analysis of a fully discrete version [SN in angle, Simple Comer Balance (SCB) in space] of the problem leads to an accurate discretized version of the continuous asymptotic diffusion problem. In other words, as ε → 0, the SCB scheme limits to an accurate discretization of the continuous asymptotic one-group diffusion equation, and the corresponding boundary conditions are also accurate. This happens even though the optical thickness of the spatial cells tends to ∞. The authors analysis shows the following: if a spatial grid is chosen that resolves the material temperature, then an accurate SCB solution is obtained in the interior of the system, even though the spatial grid is optically thick and boundary layers in the transpose solution are not resolved