Lagrange Discrete Ordinates: a new angular discretization for the three dimensional linear Boltzmann equation

TL;DR

This paper introduces the Lagrange Discrete Ordinates (LDO) equations as an alternative angular discretization method for the 3D linear Boltzmann equation, offering improved flexibility and convergence properties over classical $S_n$ methods.

Contribution

The paper derives the LDO equations based on an interpolatory framework, allowing angular flux evaluation in arbitrary directions and eliminating the need for spherical harmonic moments.

Findings

LDO equations demonstrate spectral convergence for smooth solutions.

LDO effectively mitigates ray effects with increased angular resolution.

LDO can be integrated into existing $S_n$ codes with minimal modifications.

Abstract

The classical $S_n$ equations of Carlson and Lee have been a mainstay in multi-dimensional radiation transport calculations. In this paper, an alternative to the $S_n$ equations, the "Lagrange Discrete Ordinate" (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 3D $S_n$ 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.