Accurate Least-Squares P$_N$ Scaling based on Problem Optical Thickness for solving Neutron Transport Problems

TL;DR

This paper introduces a new optical thickness-based scaling operator for the least-squares spherical harmonics method, enhancing accuracy in neutron transport problems, especially in highly scattering, optically thick media.

Contribution

A novel scaling method based on optical thickness that improves LSP_N accuracy in highly scattering media without the numerical issues of previous approaches.

Findings

Improved accuracy in optically thick, highly scattering media.

Avoids singularities present in previous reciprocal-removal scaling.

Enhances robustness of neutron transport solutions.

Abstract

In this paper, we present an accurate and robust scaling operator based on material optical thickness (OT) for the least-squares spherical harmonics (LSP$_N$) method for solving neutron transport problems. LSP$_N$ without proper scaling is known to be erroneous in highly scattering medium, if the optical thickness of the material is large. A previously presented scaling developed by Manteuffel, et al.\ does improve the accuracy of LSP$_N$, in problems where the material is optically thick. With the method, however, essentially no scaling is applied in optically thin materials, which can lead to an erroneous solution with presence of highly scattering medium. Another scaling approach, called the reciprocal-removal (RR) scaled LSP$_N$, which is equivalent to the self-adjoint angular flux (SAAF) equation, has numerical issues in highly-scattering materials due to a singular weighting. We propose a scaling based on optical thickness that improves the solution in optically thick media while avoiding the singularity in the SAAF formulation.