TY - JOUR

T1 - Iterative methods for neutron transport eigenvalue problems

AU - Scheben, Fynn

AU - Graham, Ivan G.

N1 - Preprint 08/10, Bath Institute for Complex Systems, 2010.

PY - 2011

Y1 - 2011

N2 - We discuss iterative methods for computing criticality in nuclear reactors. In general this requires the solution of a generalized eigenvalue problem for an unsymmetric integro-differential operator in six independent variables, modeling transport, scattering, and fission, where the dependent variable is the neutron angular flux. In engineering practice this problem is often solved iteratively, using some variant of the inverse power method. Because of the high dimension, matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix-free inner iterations. This leads to technically complicated inexact iterative methods, for which there appears to be no published rigorous convergence theory. For the monoenergetic homogeneous model case with isotropic scattering and vacuum boundary conditions, we show that, before discretization, the general nonsymmetric eigenproblem for the angular flux is equivalent to a certain related eigenproblem for the scalar flux, involving a symmetric positive definite weakly singular integral operator (in space only). This correspondence to a symmetric problem (in a space of reduced dimension) permits us to give a convergence theory for inexact inverse iteration and related methods. In particular this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge. We also give examples of discretizations which have a corresponding symmetric finite-dimensional reduced form. The theory is illustrated with numerical examples for several test problems of physical relevance, using GMRES as the inner solver.

AB - We discuss iterative methods for computing criticality in nuclear reactors. In general this requires the solution of a generalized eigenvalue problem for an unsymmetric integro-differential operator in six independent variables, modeling transport, scattering, and fission, where the dependent variable is the neutron angular flux. In engineering practice this problem is often solved iteratively, using some variant of the inverse power method. Because of the high dimension, matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix-free inner iterations. This leads to technically complicated inexact iterative methods, for which there appears to be no published rigorous convergence theory. For the monoenergetic homogeneous model case with isotropic scattering and vacuum boundary conditions, we show that, before discretization, the general nonsymmetric eigenproblem for the angular flux is equivalent to a certain related eigenproblem for the scalar flux, involving a symmetric positive definite weakly singular integral operator (in space only). This correspondence to a symmetric problem (in a space of reduced dimension) permits us to give a convergence theory for inexact inverse iteration and related methods. In particular this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge. We also give examples of discretizations which have a corresponding symmetric finite-dimensional reduced form. The theory is illustrated with numerical examples for several test problems of physical relevance, using GMRES as the inner solver.

UR - http://www.scopus.com/inward/record.url?scp=81555201662&partnerID=8YFLogxK

UR - http://people.bath.ac.uk/masigg/publications/preprints/SISC_includingCorrections.pdf

U2 - 10.1137/100799022

DO - 10.1137/100799022

M3 - Article

VL - 33

SP - 2785

EP - 2804

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 5

ER -