On Numerical Solutions of Transport Problems in the Diffusion Limit

A parameter ∊ is introduced into the discrete ordinates equations in such a way that as ∊ tends to zero, the solution of these equations tends to the solution of the standard diffusion equation. The behavior of spatial differencing schemes for the discrete ordinates equations is then studied, for fixed spatial and angular meshes, in the limit as ∊ tends to zero. We show that numerical solutions obtained by the diamond difference, linear characteristic, linear discontinuous, linear moments, exponential, and Alcouffe schemes all converge, in this limit, to the correct transport or diffusion result, while numerical solutions obtained by the weighted-diamond and Takeuchi schemes do not converge to the correct limiting result.