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Block Preconditioning Matrices for the Newton Method to Compute the Dominant λ-Modes Associated with the Neutron Diffusion Equation

1
Instituto Universitario de Seguridad Industrial, Radiofísica y Medioambiental, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
2
Department of Civil Environmental and Architectural Engineering, University of Padua, Via 8 Febbraio, 2, 35122 Padua, Italy
3
Department of Mathematics “Tullio Levi-Civita”, University of Padua, Via 8 Febbraio, 2, 35122 Padua, Italy
4
Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2019, 24(1), 9; https://doi.org/10.3390/mca24010009
Received: 30 November 2018 / Revised: 10 January 2019 / Accepted: 10 January 2019 / Published: 15 January 2019
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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Abstract

In nuclear engineering, the λ -modes associated with the neutron diffusion equation are applied to study the criticality of reactors and to develop modal methods for the transient analysis. The differential eigenvalue problem that needs to be solved is discretized using a finite element method, obtaining a generalized algebraic eigenvalue problem whose associated matrices are large and sparse. Then, efficient methods are needed to solve this problem. In this work, we used a block generalized Newton method implemented with a matrix-free technique that does not store all matrices explicitly. This technique reduces mainly the computational memory and, in some cases, when the assembly of the matrices is an expensive task, the computational time. The main problem is that the block Newton method requires solving linear systems, which need to be preconditioned. The construction of preconditioners such as ILU or ICC based on a fully-assembled matrix is not efficient in terms of the memory with the matrix-free implementation. As an alternative, several block preconditioners are studied that only save a few block matrices in comparison with the full problem. To test the performance of these methodologies, different reactor problems are studied. View Full-Text
Keywords: block preconditioner; generalized eigenvalue problem; neutron diffusion equation; modified block Newton method block preconditioner; generalized eigenvalue problem; neutron diffusion equation; modified block Newton method
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Carreño, A.; Bergamaschi, L.; Martinez, A.; Vidal-Ferrándiz, A.; Ginestar, D.; Verdú, G. Block Preconditioning Matrices for the Newton Method to Compute the Dominant λ-Modes Associated with the Neutron Diffusion Equation. Math. Comput. Appl. 2019, 24, 9.

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