The neutron transport equation describes the movement and interaction of neutral particles through material media. Posing the problem in spherical coordinates avoids several non-physical numerical artifacts in the simulation process, at the cost of a large increase in the computational work required to solve the resultant linear system. When discretized using traditional finite element or finite difference techniques, the first-order nature of the PDE yields linear systems of equations that are inherently serial, leading to inefficiencies when solved on large parallel architectures. While there exist very sophisticated parallel algorithms that do a good job of mitigating these inefficiencies, it is interesting to see if a more scalable algorithm can be obtained by considering non-traditional discretization techniques.
We apply the least-squares finite-element method with adaptive mesh refinement to estimate solutions to the non-scattering 1D spherical neutron transport equation. The least-squares discretization yields a symmetric positive definite linear system which shares many characteristics with systems obtained from the finite-element discretization of anisotropic elliptic PDE. Since multigrid methods applied to such systems tend to scale well on parallel architectures, the method has the potential to lead to a more efficient solution process. We demonstrate the effectiveness of the method on a mixed-media model problem and show scalable numerical results with a serial adaptive algebraic multigrid solver.