A Rayleigh Quotient Method for Criticality Eigenvalue Problems in Neutron Transport
Mario Ortega, University of California, Berkeley
Modern nuclear reactor designs require both robust and simple numerical methods. Solutions of the neutron transport equation allow a nuclear engineer to obtain physical quantities of interest. The criticality eigenvalue problems of neutron transport give insight into the time behavior of the fundamental mode neutron flux, allow for the calculation of reactor kinetics parameters and quantify how far a nuclear system is from criticality. We focus on two particular eigenvalue problems: the alpha- and k-effective eigenproblems. In general, there will be a spectrum of eigenvalues for which there are solutions. However, the only positive eigenvector corresponds to the dominant eigenvalue. Therefore, to solve these problems, it is necessary to find the dominant eigenvalue and its corresponding positive eigenvector. Traditionally, the alpha-eigenvalue problem has been solved using methods that focus on supercritical systems with large, positive eigenvalues. These methods, however, struggle for very subcritical problems where the negative eigenvalue can lead to negative absorption, potentially causing the methods to diverge. The k-effective eigenvalue problem has generally been solved using power iteration. For problems with dominance ratios close to one, however, power iteration is slow to converge and requires acceleration. We present Rayleigh quotient methods that are applied to demonstrably primitive discretizations of the neutron transport equation. We consider one-dimensional slab geometry, with diamond-difference spatial, discrete ordinates angular, and multigroup in energy discretization as our model problem. For this problem, we derive an iterative method that solves both eigenvalue problems of interest. The derived eigenvalue updates are proven to be optimal in the least squares sense and positive eigenvector updates are guaranteed from the Froebenius-Perron Theorem for primitive matrices.
Abstract Author(s): M.I. Ortega, R.N. Slaybaugh, P.N. Brown, T.S. Bailey, B. Chang