Globally Conservative, Hybrid Self-Adjoint Angular Flux and Least-Squares Method Compatible with Voids

In this paper, we derive a method for the second-order form of the transport equation that is both globally conservative and compatible with voids using the continuous finite element method. The main idea is to use the least-squares (LS) form of the transport equation in the void regions and the self-adjoint angular flux (SAAF) form elsewhere. While the SAAF formulation is globally conservative, the LS formulation needs correction in voids. The price to pay for this fix is the loss of symmetry of the bilinear form. We first derive this conservative LS (CLS) formulation in a void. Second, we combine the SAAF and CLS forms and end up with an hybrid SAAF-CLS method having the desired properties. We show that extending the theory to near-void regions is a minor complication and can be done without affecting the global conservation of the scheme. Being angular discretization-agnostic, this method can be applied to both discrete ordinates (S_N) and spherical harmonics (P_N) methods. However, since a globally conservative and void-compatible second-order form already exists for S_N [Wang et al., Nucl. Sci. Eng., Vol. 176, p. 201 (2014)] but not for P_N, we focus most of our attention on the latter angular discretization. We implement and test our method in Rattlesnake within the Multiphysics Object Oriented Simulation Environment (MOOSE) framework. The results are also compared to those of other methods.