Explicitly Restarted Arnoldi's Method for Monte Carlo Nuclear Criticality Calculations.

Abstract

A Monte Carlo implementation of explicitly restarted Arnoldi’s method is developed for estimating eigenvalues and eigenvectors of the transport-fission operator in the Boltzmann transport equation. Arnoldi’s method is an improvement over the power method which has been used for decades. Arnoldi’s method can estimate multiple eigenvalues by orthogonalising the resulting fission sources from the application of the transport-fission operator. As part of implementing Arnoldi’s method, a solution to the physically impossible—but mathematically real—negative fission sources is developed. The fission source is discretized using a first order accurate spatial approximation to allow for orthogonalization and normalization of the fission source required for Arnoldi’s method. The eigenvalue estimates from Arnoldi’s method are compared with published results for homogeneous, one-dimensional geometries, and it is found that the eigenvalue and eigenvector estimates are accurate within statistical uncertainty. The discretization of the fission sources creates an error in the eigenvalue estimates. A second order accurate spatial approximation is created to reduce the error in eigenvalue estimates. An inexact application of the transport-fission operator isalso investigated to reduce the computational expense of estimating the eigenvalues and eigenvectors. The convergence of the fission source and eigenvalue in Arnoldi’s method is analysed and compared with the power method. Arnoldi’s method is superior to the power method for convergence of the fission source and eigenvalue because both converge nearly instantly for Arnoldi’s method while the power method may require hundreds of iterations to converge. This is shown using both homogeneous and heterogeneous one-dimensional geometries with dominance ratios close to 1.