The discrete-ordinate method in diffusive regimes

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Abstract
In highly scattering regimes, the transport equation has a limit in which the leading behavior of its solution is determined by the solution of a diffusion equation. A boundary layer with a thickness of a few mean free paths usually forms between the domain boundary and the interior region, through which the given transport boundary conditions are matched to the interior tablesolution by properly choosing the boundary conditions for the diffusion equation. In order for a numerical scheme to be effective in these regimes, it must have both a correct interior diffusion limit and a correct boundary condition limit. The behavior of the discrete-ordinate method is studied in these limits and formulas for the resulting diffusion equation and its boundary conditions are found. By imposing the condition that these limiting formulas be the same as those for the transport equation, a new constraint on the quadrature set is obtained and a way to discretize the boundary data based on the discrete W-function is suggested. Numerical examples for isotropic materials demonstrate our theoretic results.