Improved Fully-Implicit Spherical Harmonics Methods for First and Second Order Forms of the Transport Equation Using Galerkin Finite Element

In this dissertation, we focus on solving the linear Boltzmann equation -- or transport equation -- using spherical harmonics (PN) expansions with fully-implicit time-integration schemes and Galerkin Finite Element spatial discretizations within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. The presentation is composed of two main ensembles. On one hand, we study the first-order form of the transport equation in the context of Thermal Radiation Transport (TRT). This nonlinear application physically necessitates to maintain a positive material temperature while the PN approximation tends to create oscillations and negativity in the solution. To mitigate these flaws, we provide a fully-implicit implementation of the Filtered PN (FPN) method and investigate local filtering strategies. After analyzing its effect on the conditioning of the system and showing that it improves the convergence properties of the iterative solver, we numerically investigate the error estimates derived in the linear setting and observe that they hold in the non-linear case. Then, we illustrate the benefits of the method on a standard test problem and compare it with implicit Monte Carlo (IMC) simulations. On the other hand, we focus on second-order forms of the transport equation for neutronics applications. We mostly consider the Self-Adjoint Angular Flux (SAAF) and Least-Squares (LS) formulations, the former being globally conservative but void incompatible and the latter having -- in all generality -- the opposite properties. We study the relationship between these two methods based on the weakly-imposed LS boundary conditions. Equivalences between various parity-based PN methods are also established, in particular showing that second-order filters are not an appropriate fix to retrieve void compatibility. The importance of global conservation is highlighted on a heterogeneous multigroup k-eigenvalue test problem. Based on these considerations, we propose a new method that is both globally conservative and compatible with voids. The main idea is to solve the LS form in the void regions and the SAAF form elsewhere. For the LS form to be conservative in void, a non-symmetric fix is required, yielding the Conservative LS (CLS) formulation. From there, a hybrid SAAF-- CLS method can be derived, having the desired properties. We also show how to extend it to near-void regions and time-dependent problems. While such a second-order form already existed for discrete-ordinates (SN) discretizations (Wang et al. 2014), we believe that this method is the first of its kind, being well-suited to both SN and PN discretizations