ITERATIVE METHODS FOR NEUTRON TRANSPORT EIGENVALUE PROBLEMS

被引:9
作者
Scheben, Fynn [1 ]
Graham, Ivan G. [1 ]
机构
[1] Univ Bath, Dept Math Sci, Bath BA2 7AY, Avon, England
基金
美国国家科学基金会; 英国工程与自然科学研究理事会;
关键词
neutron transport; criticality; generalized eigenvalue problem; symmetry; inexact inverse iteration; INEXACT INVERSE ITERATION; EQUATION; CONVERGENCE; SOLVES;
D O I
10.1137/100799022
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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.
引用
收藏
页码:2785 / 2804
页数:20
相关论文
共 50 条
[41]   A POD reduced-order model for resolving the neutron transport problems of nuclear reactor [J].
Sun, Yue ;
Yang, Junhe ;
Wang, Yahui ;
Li, Zhuo ;
Ma, Yu .
ANNALS OF NUCLEAR ENERGY, 2020, 149
[42]   Iterative Krylov Methods for Acoustic Problems on Graphics Processing Unit [J].
Ahamed, Abal-Kassim Cheik ;
Magoules, Frederic .
PROCEEDINGS OF THIRTEENTH INTERNATIONAL SYMPOSIUM ON DISTRIBUTED COMPUTING AND APPLICATIONS TO BUSINESS, ENGINEERING AND SCIENCE, (DCABES 2014), 2014, :19-23
[43]   Iterative methods for solving nonlinear problems of nuclear reactor criticality [J].
A. M. Kuz’min .
Physics of Atomic Nuclei, 2012, 75 :1551-1556
[44]   Iterative Methods for Solving Nonlinear Problems of Nuclear Reactor Criticality [J].
Kuz'min, A. M. .
PHYSICS OF ATOMIC NUCLEI, 2012, 75 (13) :1551-1556
[45]   AOR type iterative methods for solving least squares problems [J].
Wang, L .
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS, 2001, 77 (01) :105-116
[46]   Design and multidimensional extension of iterative methods for solving nonlinear problems [J].
Artidiello, S. ;
Cordero, Alicia ;
Torregrosa, Juan R. ;
Vassileva, M. P. .
APPLIED MATHEMATICS AND COMPUTATION, 2017, 293 :194-203
[47]   Iterative Methods for Solving Split Feasibility Problems and Fixed Point Problems in Banach Spaces [J].
Liu, Min ;
Chang, Shih-sen ;
Zuo, Ping ;
Li, Xiaorong .
FILOMAT, 2019, 33 (16) :5345-5353
[48]   ITERATIVE METHODS WITH ANALYTICAL PRECONDITIONING TECHNIQUE TO LINEAR COMPLEMENTARITY PROBLEMS: APPLICATION TO OBSTACLE PROBLEMS [J].
Najafi, H. Saberi ;
Edalatpanah, S. A. .
RAIRO-OPERATIONS RESEARCH, 2013, 47 (01) :59-71
[49]   Product eigenvalue problems [J].
Watkins, DS .
SIAM REVIEW, 2005, 47 (01) :3-40
[50]   A-posteriori residual bounds for Arnoldi's methods for nonsymmetric eigenvalue problems [J].
Dookhitram, Kumar ;
Boojhawon, Ravindra ;
Gopaul, Ashvin ;
Bhuruth, Muddun .
NUMERICAL ALGORITHMS, 2011, 56 (04) :481-495