Discrete ordinates quadrature schemes based on the angular interpolation of radiation intensity

The discrete ordinates method fails in treating specular reflection at the boundary because the quadratures on the sphere do not assume any analytic r epresentation of a function under integration and, therefore, the intensity of the specularly reflected beam is undetermined. To solve this problem a new approach to the construction of quadrature schemes is presented. The ap proach is based on the quasilinear angular interpolation of radiation inten sity given at the nodal points of a triangular grid on the sphere. Two type s of triangulation on the sphere and two kinds of quasilinear interpolation are described. Triangulations considered are invariant with respect to the octahedron symmetry group. As a consequence, quadrature sets on the sphere obtained by this approach are also invariant with respect to the group. It is shown that the arrangement of the nodal points crucially affects the ac curacy of the quadratures and arrangements which provide high accuracy of t he quadratures are chosen. Due to the analytical representation of the angu lar dependence of radiation intensity the treatment of specular reflective boundaries becomes straightforward. (C) 2001 Elsevier Science Ltd. All righ ts reserved.