Conservation of scattered energy and asymmetry factor in the new Rotationally Symmetric Spherical Discretisation scheme

Abstract

A Rotationally Symmetric Spherical Discretisation (RSSD) technique has recently been developed for the discretisation of scattering phase functions in the Discrete Ordinates Method (DOM) in a consistent and rational manner. RSSD has inherent energy conservation, is suitable for any quadrature scheme used, minimising run-time matrix calculation, and to date has been used for the S4, S6 and S8 ordinate sets. In this follow-up paper, the RSSD boundary intervals for the four weightings of the S10 ordinate set are presented. The effect of asymmetry factor g and angular resolution of the discrete input scattering phase function distribution “grid” on final RSSD errors (in scattered energy conservation and calculated asymmetry factor) is explored, making use of the Henyey–Greenstein (HG) family of distributions. It is shown that the RSSD scattered energy error ξeRSSD is a strong function of the grid resolution, indicating that ξeRSSD can be controlled down to user-required levels by judicious choice of grid resolution and by local grid refinement in regions of steep gradients and peaks in the scattering phase function distribution (in this paper error levels as low as 0.002% have been achieved). RSSD asymmetry factor error ξgRSSD behaves differently to ξeRSSD, however, displaying a strong grid resolution dependence in a “grid sensitivity zone” of combinations of coarse grid resolution and high values of prescribed g (for the HG family of scattering phase functions, above g values of 0.9, 0.95 and 0.98 for grid angular resolutions of 1°, 0.5° and 0.25° respectively). Outside of the “grid sensitivity zone” g error is insensitive to grid resolution and achieves maximum (positive and negative) values of between −2.7% and 1% between prescribed g values of 0.7 and 0.9. This band decreases to a range from −1% to 1% for the S4, S6 and S10 ordinate sets. The grid resolution conclusions are supported by examining RSSD discretisation for two real scattering phase function distributions: for a diffusely reflecting large sphere with an asymmetry factor of −0.4444, grid angular resolutions of 2°, 1°, 0.5° and 0.25° resulted in the decreasing maximum ξeRSSD values of 0.0449%, 0.0136%, −0.0050% and −0.0015% respectively, but near-constant maximum ξgRSSD values of −1.225%, −1.239%, −1.247% and −1.250% respectively, while for a transparent large sphere with an asymmetry factor of 0.660, grid angular resolutions of 1°, 0.5° and composite 0.5°/0.25° resolution resulted in the decreasing maximum ξeRSSD values of 0.0685%, −0.0402% and −0.0046% respectively and near-constant maximum ξgRSSD values of −1.286%, −1.310% and −1.313% respectively.