Uniform Positive-Weight Quadratures for Discrete Ordinate Transport Calculations

Abstract

Mechanical quadratures that allow systematic improvement and solution convergence are derived for application of the discrete ordinates method to the Boltzmann transport equation. The quadrature directions are arranged on n latitudinal levels, are uniformly distributed over the unit sphere, and have positive weights. Both a uniform and equal-weight quadrature set UEn and a uniform and Gauss-weight quadrature set UGn are derived. These quadratures have the advantage over the standard level-symmetric LQn quadrature sets in that the weights are positive for all orders, and the solution may be systematically converged by increasing the order of the quadrature set. As the order of the quadrature is increased the points approach a uniform continuous distribution on the unit sphere and the quadrature is invariant with respect to spatial rotations. The numerical integrals converge for continuous functions as the order of the quadrature is increased.Numerical calculations were performed to evaluate the application of the UEn quadrature set. Comparisons of the exact moments and those calculated using the UEn quadrature set demonstrate that the moment integrals are performed accurately except for distributions that are very sharply peaked along the direction of the polar axis. A series of DORT transport calculations of the >1-MeV neutron flux for a typical reactor core/pressure vessel geometry were also carried out. These calculations employed the UEn (n = 6, 10, 12, 18, and 24) quadratures and indicate that the UEn solutions have converged to within ~0.5%. The UE24 solutions were also found to be more accurate than the calculations performed with the S16 level-symmetric quadratures.