Remedies for Ray Effects

Remedies are considered for the ray effect, a flux distortion producing defect of the discrete ordinates approximation to the transport equation. The partially effective remedies of increasing the number of directions, selecting quadrature sets invariant under discrete rotations, and introducing coupling terms into the representation of the transport divergence operator are considered. Numerical results are presented showing the effectiveness of these remedies in a standard test problem. Remedies that eliminate the ray effect are also considered. We show how to formulate multidimensional discrete ordinate equations equivalent to spherical harmonic equations and relate the order of the equivalent spherical harmonic equations to the degree of precision of the numerical quadrature. We give numerical results for this kind of remedy in three test problems, including one with material discontinuities. We show how the use of orthonormal polynomials permits the formulation of discrete ordinates approximations which have properties “like” those of the highest order spherical harmonic equations consistent with the number of directions in the discrete ordinates approximation. We find that these formulations also eliminate ray effects. We conclude that the partially effective remedy of using more, specially chosen, directions is the most practical remedy for most applications, but that for especially difficult situations or for reference calculations, the defect-eliminating spherical harmonic-like formulations should be available for use.