This thesis focuses on iterative methods for the treatment of the steady state neutron transport equation in slab geometry, bounded convex domain of Rn (n = 2,3) and in 1-D spherical geometry. We introduce a generic Alternate Direction Implicit (ADI)-like iterative method based on positive definite and m-accretive splitting (PAS) for linear operator equations with operators admitting such splitting. This method converges unconditionally and its SOR acceleration yields convergence results similar to those obtained in presence of finite dimensional systems with matrices possessing the Young property A. The proposed methods are illustrated by a numerical example in which an integro-differential problem of transport theory is considered. In the particular case where the positive definite part of the linear equation operator is self-adjoint, an upper bound for the contraction factor of the iterative method, which depends solely on the spectrum of the self-adjoint part is derived. As such, this method has been successfully applied to the neutron transport equation in slab and 2-D cartesian geometry and in 1-D spherical geometry. The self-adjoint and m-accretive splitting leads to a fixed point problem where the operator is a 2 by 2 matrix of operators. An infinite dimensional adaptation of minimal residual and preconditioned minimal residual algorithms using Gauss-Seidel, symmetric Gauss-Seidel and polynomial preconditioning are then applied to solve the matrix operator equation. Theoretical analysis shows that the methods converge unconditionally and upper bounds of the rate of residual decreasing which depend solely on the spectrum of the self-adjoint part of the operator are derived. The convergence of theses solvers is illustrated numerically on a sample neutron transport problem in 2-D geometry. Various test cases, including pure scattering and optically thick domains are considered.

Two new concepts have been explored in solving the neutron diffusion equation in one and two dimensions. At the present time, the diffusion equation is solved using source iterations. These iterations are performed in a mathematical form which has a great deal of physical significance. Specifically, the neutron production term is on the right-hand side, while the absorption and leakage terms are on the left side. In performing a single source iteration, a distribution for the neutron flux is assumed so that the production term can be calculated. This provides a "known" right-hand side. Solving the difference equation for the flux, which corresponds to this assumed source distribution, gives the next estimate for the flux distribution. This type of iteration has the physically significant characteristic of finding directly, for each iteration, a flux which corresponds to an assumed source distribution. In this thesis it was found that by subtracting the absorption term from both sides of the diffusion equation, and performing "source iterations" with both absorption and production terms on the right-hand side (and only the leakage term on the left-hand side), improved convergence rates were attained in many cases. In one neutron energy group, this new idea of putting the absorption term on the right-hand side worked best with only one region, and where reactor dimensions were large compared to the thermal neutron diffusion length (a”L). In small reactors, where a=L, convergence behavior was similar for both forms of iteration. This new idea was also found to work quite well in one-group multiregion problems. However, due to problems with numerics (inherent asymmetric treatment of the scattering terms), the method does not work at all in a multi-energy group formulation. Secondly, in two dimensions, a closed-form solution to a single source iteration has been found. At this time, the standard method of solution for a two-dimensional source iteration is to perform "inner iterations" to approximately solve for the flux that corresponds to an assumed source. The alternative, up until now, was to solve a giant matrix of the order (N2 x N2). This is a sparse matrix, but it has always been considered as highly undesirable to work with a solution (even though it may be closed-form) where the matrix to be solved increases in order roughly as the fourth power of the number of mesh intervals. The new algebraic form for this closed-form solution involves a matrix of order (N x N), not (N2 x N2). The matrix is, however, a full matrix. What is done, essentially, is to solve simultaneously for all the flux values along the vertical centerline of the two-dimensional problem, and then use a reflective boundary condition across the core centerline, and then the difference equation itself (in vector form) as a set of flux-vector generating equations to generate the entire flux field, line by line. In solving for the first flux vector (at the x = o, or z = o, core centerline), the right-hand side of the matrix problem incorporates all of the source values in the entire problem space. The initial inversion of the full (N x N) matrix algebraically guarantees that the (M+1)th flux vector (on the problem space boundary) will go to zero. This matrix method for two-dimensional neutronic analysis was shown to work well in both cartesian and cylindrical coordinates.

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.