Variational solutions to the neutron transport equation in P[subscript N] approximation

Abstract

A new numerical method is developed to solve the multigroup steady state neutron transport equation in P(,N) approximation in one dimensional slab and spherical geometry. The multigroup P(,N) moment equations are transformed into coupled elliptic-like differential equations by defining a particular new function which is a linear combination of the odd angular moments of the neutron flux. The relationship between this new function and the neutron flux angular moments is determined. The neutron transport flux interface boundary conditions and the Mark and Marshak vacuum boundary conditions are derived in terms of the new function. The resultant coupled differential equations are solved using a variational method with cubic shape functions that satisfy appropriate transport continuity and jump conditions at node and material interface boundaries. Additionally, Lagrangian constraints on the functional are applied to guarantee exact neutron conservation of the numerical method. Spatially dependent material properties are approximated by cubic splines. Both external sources and eigenvalue problems (multiplication factor and critical dimensions) are addressed. Anisotropic scattering and upscattering in energy are available in the multigroup scattering approximation. In order to validate the numerical computer program based on the above methodology, a number of new analytical benchmark solutions were precisely constructed to the P(,N) equations for both eigenvalues and fluxes. The one group P(,N) solutions are obtained for one material slab and sphere critical dimension problems parametrically for the excess number of neutrons per collision (1.02 (LESSTHEQ) C (LESSTHEQ) N 2.0) and various N (1 (LESSTHEQ) N (LESSTHEQ) 11). The P(,3) six group Lady Godiva spherical critical benchmark problem is solved for multiplication factor, critical dimensions, and fluxes using both Mark and Marshak vacuum boundary conditions. The two material region two group P(,3) slab geometry (cell problem) is solved for the multiplication factor and the fluxes. These same problems were solved using the conservative variational method developed herein. The results of the numerical variational method are compared in detail with the analytical P(,N) solutions and with other existing benchmark solutions. A two region two group external source problem is solved to investigate the validity of the superposition principle in the framework of the numerical transport theory program.