A block Arnoldi method for the SP_N equations

ABSTRACT

The simplified spherical harmonics equations are a useful approximation to the stationary neutron transport equation. The eigenvalue problem associated with them is a challenging problem from the computational point of view. In this work, we take advantage of the block structure of the involved matrices to propose the block inverse-free preconditioned Arnoldi method as an efficient method to solve this eigenvalue problem. For the spatial discretization, a continuous Galerkin finite element method implemented with a matrix-free technique is used to keep reasonable memory demands. A multilevel initialization using linear shape functions in the finite element method is proposed to improve the method convergence. This initialization only takes a small percentage of the total computational time. The proposed eigenvalue solver is compared to the standard power iteration method, the Krylov-Schur method and the generalized Davidson method. The numerical results show that it reduces the computational time to solve the eigenvalue problem.

KEYWORDS:

• Generalized eigenvalue problem
• neutron transport
• multilevel method
• block inverse-free preconditioned Arnoldi method
• generalized Davidson method

2010 MSC MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65N30
• 65N25
• 35J47

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

A. Vidal-Ferràndiz http://orcid.org/0000-0001-5449-7356

A. Carreño http://orcid.org/0000-0003-2302-1157

D. Ginestar http://orcid.org/0000-0003-1243-6648

G. Verdú http://orcid.org/0000-0001-5098-080X

Additional information

Funding

This work has been partially supported by Spanish Ministerio de Economía y Competitividad under projects ENE2017-89029-P, MTM2017-85669-P and BES-2015-072901. Moreover, it has been supported by the Cátedra of the CSN Vicente Serradell.