Progress in nodal methods for the solution of the neutron diffusion and transport equations

Abstract

Recent progress in the development of coarse-mesh nodal methods for the numerical solution of the neutron diffusion and transport equations is reviewed. In contrast with earlier nodal simulators, more recent nodal diffusion methods are characterized by the systematic derivation of spatial coupling relationships that are entirely consistent with the multigroup diffusion equation. These relationships most often are derived by developing approximations to the one-dimensional equations obtained by integrating the multidimensional diffusion equation over directions transverse to each coordinate axis. Both polynomial and analytic approaches to the solution of the transverse-integrated equations are discussed, and the Cartesian-geometry polynomial approach is derived in a manner which motivates the extension of this formulation to the solution of the diffusion equation in hexagonal geometry. Iterative procedures developed for the solution of the nodal equations are discussed briefly, and numerical comparisons for representative three-dimensional benchmark problems are given. The application of similar ideas to the neutron transport equation has led to the development of coarse-mesh transport schemes that combine nodal spatial approximations with angular representations based on either the standard discrete-ordinate approximation or double P n expansions of the angular dependence of the fluxes on the surfaces of the nodes. The former methods yield improved difference approximations to the multidimensional discrete-ordinates equations, while the latter approach leads to equations similar to those obtained in interface-current nodal-diffusion formulations. The relative efficiencies of these two approaches are discussed, and directions for future work are indicated.