# Least-Squares Finite Element Discretization of the Neutron Transport Equation in Spherical Geometry

- 2015

#### Abstract

The main focus of this paper is the numerical solution of the Boltzmann transport equation for neutral particles through mixed material media in a spherically symmetric geometry. Standard solution strategies, like the Discrete Ordinates Method (DOM), may lead to nonphysical approximate solutions. In particular, a point source at the center of the sphere yields undesirable ray effects. Posing the problem in spherical coordinates avoids ray effects and other non-physical numerical artifacts in the simulation process, at the cost of coupling all angles in the PDE setting. In addition, traditional finite element or finite difference techniques for spherical coordinates often yield incorrect scalar flux at the center of the sphere, known as flux dip, and oscillations near steep gradients. In this paper, a least-squares finite element method with adaptive mesh refinement is used to approximate solutions to the non-scattering one-dimensional neutron transport equation in spherically symmetric geometry. It is shown that the resulting numerical approximations avoid flux dip and oscillations. The least-squares discretization yields a symmetric positive definite linear system which shares many characteristics with systems obtained from Galerkin finite element discretization of totally anisotropic elliptic PDEs. In general, standard Algebraic Multigrid (AMG) techniques fail to scale on non-grid-aligned anisotropies. In this paper, a new variation of Smoothed Aggregation (SA) is employed and shown to be essentially scalable. The effectiveness of the method is demonstrated on several mixed-media model problems. 1. Introduction. The main focus of this paper is the numerical solution of the Boltzmann transport equation for neutral particles in a spherically symmetric domain with mixed media. A First-Order System Least-Squares (FOSLS) finite element dis-cretization is developed and shown to be accurate and to avoid flux dip near the center of the sphere and oscillations near steep fronts caused by material boundaries. The least-squares functional provides a locally sharp and globally accurate a posteriori error measure which is used in conjunction with an adaptive mesh strategy, ACE, that is based on Accuracy per Computational cost [7, 5]. FOSLS discretization of the non-scattering transport operator in spherically symmetric geometry yields a linear system that resembles an anisotropic diffusion operator and is, therefore, amenable to solution by a recently developed Smoothed Aggregation (SA) [25] algorithm specifically aimed at anisotropic diffusion problems [22]. In this context, the SA algorithm is shown to converge independently of the number of refinement levels. This paper will demonstrate that the FOSLS-ACE-SA approach provides an accurate and very …

**DOI:**10.1137/140975152