Analysis of Projective-Iterative Methods for Solving Multidimensional Transport Problems

Abstract

The particle transport equation has a wide range of applications: nuclear engineering, astrophysics, atmospheric science, medical physics, microelectronics manufacturing, etc. It is an integro-differential equation with seven independent variables: 3 spatial, 2 angular, energy, and time, which cannot be solved analytically in most of the cases of interest. The way to solve this equation is to discretize it in space, angle, energy, and time. In practical cases, this leads to a huge sparse matrix. Iterative methods should be used even for solving transport problems on the most powerful computers available nowadays. The need to analyze the behavior of these methods is obvious: knowledge about behavior of methods can help us to improve them and avoid their use in cases in which they are not efficient. Also, if we can predict what should happen in specific cases, we can verify and validate transport codes. Analysis of iterative methods' behavior in highly scattering and strong heterogeneous medium is very important from the point of view of solving various radiative and particle transport problems. It became important for solving neutron transport equation in full-core, due to current industry's interest in obtaining very detailed transport solution without homogenization. For these reasons, the main target of this thesis was to analyze the convergence rate of four methods used to solve the steady state transport equation. We were interested in studying behavior of these methods in case of one and two dimensional strong heterogeneous and highly scattering medium with periodic structure, on rectangular grids. In order to understand better these methods, we analyzed them as well in cases of homogeneous and low scattering medium, uniform grids, etc. The main tool that we used is Fourier analysis. Iteration matrix analysis was a secondary tool that we consider. It proved to be restrictive in some cases but provided a good insight of the methods behavior. In several diffcult cases the Fourier analysis predicted degradation in efficiency or even divergence for the methods that we've studied. In most of the cases, the numerical results were consistent with the analytic predictions. In order to cover various areas where the transport equation is used, we spanned wide ranges for parameters of transport equation. Most of the cases in which the considered methods demonstrate slow convergence or even divergence are not specific to nuclear reactors. It means that one can apply these methods for solving reactor-physics problems.