Accurate Least-Squares P N Scaling based on Problem Optical Thickness for solving Neutron Transport Problems Weixiong Zheng a,∗ , Ryan G. McClarren b a Department of Nuclear Engineering, University of California, Berkeley, 4103 Etcheverry Hall, Berkeley, CA 94720 b Texas A&M Nuclear Engineering, 3133 TAMU, College Station, TX 77843-3133 Abstract In this paper, we present an accurate and robust scaling operator based on ma- terial 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 that method, however, essentially no scaling is applied in optically thin materials, which can lead to an erroneous solutions in the presence of a 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 im- proves the solution in optically thick media while avoiding the singularity in the SAAF formulation. Keywords: Least Square P N , Neutron Transport Equation, Reactor Shielding, Optical Thickness, Scaling, Thick Diffusion Limit * Corresponding author Email addresses: zwxne2010@gmail.com (Weixiong Zheng), rgm@tamu.edu (Ryan G. McClarren) Preprint submitted to Progress of Nuclear Energy October 19, 2018 arXiv:1609.06699v2 [physics.comp-ph] 15 May 2017