Ray Effects in Discrete Ordinates Equations

The nature of anomalous computational effects due to the discretization of the angular variable in transport theory discrete ordinates approximations is described and analyzed. The origin of these effects within the derivation of the Sn discrete ordinates equations is shown, and the effects are related to the non-equivalence of the general geometry discrete ordinates equations and the corresponding spherical harmonics equations. Procedures are given for the definition of two-dimensional discrete ordinates equations that are equivalent to the spherical harmonics equations. Elimination of ray effects from the two-dimensional S2 equations by reduction to the diffusion theory equations is verified in a numerical example. Recipes for the elimination of ray effects are analyzed in the analytic solution of the infinite medium, isotropic line-source problem in the rectangular geometry, S2 approximation. Optimum magnitudes for corrective source terms are indicated by the analysis. It is concluded that ray effects may be eliminated by modification of the discrete ordinates formulation, but that the extra computational effort may be more expensive than the alternative of increasing the order of angular quadrature and that the presence of discretization effects may serve as an indicator of the adequacy of the angular quadrature used.