Moments-Based Versions of the Spherical Geometry Method of Tubes in Thick, Diffusive Regions

Characteristic methods are widely known to be very accurate approaches to the solution of numerical transport problems. These methods are most often used for neutron transport applications (i.e., lattice physics calculations) where spatial cells are of intermediate optical thickness [O(1) to O(100) mean free paths, depending on the energy group] and materials are not exceptionally highly scattering (scattering ratios < 0.999). There has been interest in using characteristic methods for radiative transfer applications, which often involve very optically thick and diffusive regions. Previous work has involved analyses of families of Cartesian geometry characteristic methods in optically thick and diffusive regions. There is a significant body of work in the Russian literature on curvilinear geometry characteristic methods, but very few analyses of their behavior in thick diffusive regions have been published. In this paper we develop two new members of a family of one-dimensional spherical geometry characteristic methods - the method of tubes. These new methods are similar to traditional slab geometry characteristics methods in that they utilize spatial moments of the transport equation in each cell to generate the data used in the representation of the total source (scattering source plus external source). We present the results of an asymptotic analysis of these methods to predict their behavior in the thick diffusion limit, and we compare these predictions with numerical results from several test problems. This analysis shows that the constant source (step) method behaves very poorly in the diffusion limit, but that the linear source method is accurate in this physical regime.