Lagrange Discrete Ordinates: A New Angular Discretization for the Three-Dimensional Linear Boltzmann Equation

The classical S_n equations of Carlson and Lee have been a mainstay in multidimensional radiation transport calculations. In this paper, an alternative to the S_n equations, the “Lagrange Discrete Ordinates” (LDO) equations, are derived. These equations are based on an interpolatory framework for functions on the unit sphere in three dimensions. While the LDO equations retain the formal structure of the classical S_n equations, they have a number of important differences. The LDO equations naturally allow the angular flux to be evaluated in directions other than those found in the quadrature set. To calculate the scattering source in the LDO equations, no spherical harmonic moments are needed—only values of the angular flux. Moreover, the LDO scattering source preserves the eigenstructure of the continuous scattering operator. The formal similarity of the LDO equations with the S_n equations should allow easy modification of mature three-dimensional S_n codes such as PARTISN or PENTRAN to solve the LDO equations. Numerical results are shown that demonstrate the spectral convergence (in angle) of the LDO equations for smooth solutions and the ability to mitigate ray effects by increasing the angular resolution of the LDO equations.