Improved negative fixup with cell rebalance for boltzmann-fokker-planck transport equation

Abstract

The transport of neutrons is usually governed by the Boltzmann equation. The scattering cross sections are expanded into Legendre polynomial series. This assumes that scattering is almost isotropic. If scattering is highly anisotropic, this expansion is unsuitable. So, the Boltzmann-Fokker-Planck (BFP) equation was proposed previously, which combines the advantages of the usual transport equation and the Fokker-Planck equation. Because the BFP equation involves angular flux derivatives with respect to energy and direction, the standard neutron transport codes cannot be used to solve the BFP equation. Diamond difference scheme can be applied to the BFP equation with respect to energy and space. This scheme is accurate for finite mesh size, but it is not absolutely safe to use. If negative flux comes out of one mesh, then the solution can be completely wrong. So, diamond difference scheme needs a negative flux fixup, and comservative fixup was proposed by Przybylski and Liguo. In this study, a negative flux fixup scheme with cell-rebalanced average flux is developed. Because negative flux has influence on solution, to avoid this fault, the nodal method (constant-constant) is also applied to the BFP eqaution with respect to energy and space. Since the nodal method is a more positvie scheme than diamond difference, it can be an alternative for the BFP equation.