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Iterative methods for neutron transport eigenvalue problems


Reference:

Scheben, F. and Graham, I. G., 2011. Iterative methods for neutron transport eigenvalue problems. SIAM Journal on Scientific Computing, 33 (5), pp. 2785-2804.

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    Official URL:

    http://dx.doi.org/10.1137/100799022

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    Abstract

    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.

    Details

    Item Type Articles
    CreatorsScheben, F.and Graham, I. G.
    DOI10.1137/100799022
    Related URLs
    URLURL Type
    http://people.bath.ac.uk/masigg/publications/preprints/SISC_includingCorrections.pdfAuthor
    DepartmentsFaculty of Science > Mathematical Sciences
    Research CentresBath Institute for Complex Systems (BICS)
    Publisher StatementGraham_Scheben_2011.pdf: ©SIAM
    RefereedYes
    StatusPublished
    ID Code26696
    Additional InformationPreprint 08/10, Bath Institute for Complex Systems, 2010.

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