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.Ph.D.Nuclear Engineering & Radiological SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64765/1/jlconlin_1.pd
