Spherical Harmonics Finite Element Transport Equation Solution Using a Least-Squares Approach

To mitigate some drawbacks of the discrete ordinates method or the even-parity approach, a new deterministic method for solving the Boltzmann transport equation is proposed. Based on a scaled least-squares formulation, the first-order transport equation is solved for a spherical harmonics expansion of the angular flux. This approach allows a continuous finite element discretization. Discrete equations have been derived for media with anisotropic scattering. Moreover, extensions are proposed to allow for solutions in three-dimensional multiplicative regions. Asymptotic analyses of this least-squares approach show the need for a scaling of the transport equation in order to maintain the diffusion limit. One-dimensional tests are used to evaluate this scaling operator, and results are compared with reference solutions. Anisotropic multigroup scattering cases are also presented. Tests on a three-dimensional simple problem show that ARTEMIS, the transport solver based on this method, gives solutions free of ray effects.