Approximate solution to neutron transport equation using spherical harmonics expansion and a conservative variational principle

Abstract

An approximate solution method has been developed to solve the one-dimensional, steady-state neutron transport problems in plane and spherical geometries. The spherical harmonics expansion and the multigroup approximation are employed to represent the angular- and energy-dependence. The angular moments are replaced by a set of transformation functions that leads to the second-order form of the multigroup P(,N) equations. The approximate solutions of the transformation functions are formulated by a variational principle in conjunction with the cubic Hermite polynomials. Conservation constraints are imposed by the usage of Lagrange multipliers. In order to validate the numerical solutions, the analytical expressions of criticality conditions and angular moments to the multigroup P(,N) equations are constructed by applying the eigenfunction expansion technique. This analytical approach is further extended for problems in cylindrical geometry. In this study, both external sources and criticality problems are addressed. Accuracy and reliability of the approximate solution methods are investigated by comparing with the benchmark calculations or other conventional methods. Preliminary results are reported, and recommendations for future research are made.