Analysis of the method of tubes characteristic schemes in the thick diffusive limit

Graduation date: 2009Characteristic methods are widely known to be very accurate approaches to the\ud solution of numerical transport problems. These methods are most often used for\ud neutron transport applications (i.e. lattice physics calculations) where spatial cells\ud are of intermediate optical thickness (O(1) - O(100) mean free paths, depending on\ud the energy group) and materials are not exceptionally highly scattering (scattering\ud ratios < 0.999). There has been interest in using characteristic methods for radiative\ud transfer applications, which often involve very optically thick and diffusive\ud regions. Previous work has involved analyses of families of Cartesian geometry\ud characteristic methods in optically thick and diffusive regions. There is a significant\ud body of work in the Russian literature on curvilinear geometry characteristic\ud methods, but very few analyses of their behavior in thick diffusive regions have\ud been published. This thesis will focus on the diffusion limit of a specific family of\ud 1-D spherical geometry characteristic methods - the method of tubes (MOT). In\ud these methods, the streaming operator is transformed via a change of coordinates\ud into a slab-geometry-like form.\ud First we present two MOT discretizations published in the Russian literature\ud and two new variants (SC, LC) based on traditional slab geometry characteristic\ud approaches. We have performed asymptotic analyses of these four characteristic\ud methods and have verified these analyses with numerical results.\ud Optically thick and diffusive problems are often very computationally expensive\ud to solve using the traditional "source iteration" iterative method. We have developed an efficient acceleration technique for the most promising version of the MOT (LC) based on an approach developed for the slab geometry linear discontinuous\ud finite element method. The LC version of the MOT that we have developed has very good thick diffusion limit behavior and many other very attractive properties