Modified least-squares finite element method for solving the Boltzmann transport equation with spherical harmonic angular approximation

This paper presents a modified least-squares finite element method for solving the Boltzmann transport equation with spherical harmonics angular approximation (LSP_N). Traditional LSP_N methods usually adopt weakly imposed boundary conditions which contain an uncertain parameter and it may lead to inaccurate results without proper scaling for problems with sharp cross section interfaces. In order to eliminate the discretization error at material interface and use continuous interface condition, we multiply a scaling parameter to least-squares term and add the weak form of the first-order transport equation. With this method, the natural boundary condition can be applied directly and the scaling parameter is identified as the minimal value of characteristic length of element and the reciprocal of removal cross section. The method is verified by various benchmark problems and the numerical results show it has high accuracy, stability and void applicability.