A Krylov–Schur solution of the eigenvalue problem for the neutron diffusion equation discretized with the Raviart–Thomas method

Abstract

Mixed-dual formulations of the finite element method were successfully applied to the neutron diffusion equation, such as the Raviart-Thomas method in Cartesian geometry and the Raviart-Thomas-Schneider in hexagonal geometry. Both methods obtain system matrices which are suitable for solving the eigenvalue problem with the preconditioned power method. This method is very fast and optimized, but only for the calculation of the fundamental mode. However, the determination of non-fundamental modes is important for modal analysis, instabilities and fluctuations of nuclear reactors. So, effective and fast methods are required for solving eigenvalue problems.The most effective methods are those based on Krylov subspaces projection combined with restart, such as Krylov-Schur.In this work, a Krylov-Schur method has been applied to the neutron diffusion equation, discretized with the Raviart-Thomas and Raviart-Thomas-Schneider methods.

Multigroup neutron diffusion equation with the finite volume method in reactors using MOX fuels

“…On the other hand, for the PWR MOX/UO 2 Benchmark, the reference solution was obtained with TRIVAC code [17], which is a diffusion code. TRIVAC can solve the neutron diffusion equation with several methods, but in this work the authors used the authors used a version of TRIVAC based on its version 5, but including an eigensolver based on the SLEPc library [19].…”

Block hybrid multilevel method to compute the dominant λ-modes of the neutron diffusion equation

“…This problem can be mitigated using the Jacobi-Davidson method, (Verdú et al, 2005), that also makes use of a shift and invert strategy, but it does not need to solve as many linear systems as the previous ones. Moreover, classical methods such as the shifted inverse iteration method for the computation of several eigenvalues make use of a deflation process and this has been shown to have a slow converge when it is compared with Krylov-Schur method (Bernal et al, 2017).…”

Development of a 3D Modal Neutron Code with the Finite Volume Method for the Diffusion and Discrete Ordinates Transport Equations. Application to Nuclear Safety Analyses

“…Actually, the original version uses the Hotelling Deflation technique to do so, which is not the most efficient approach. However, the author of this thesis developed an algorithm for applying the Krylov-Schur method of SLEPc in TRIVAC, which was published in Bernal et al 2017a. This Krylov-Schur method is very efficient for calculating several eigenvalues.…”

“…In the last years, a number of works has been published, which use Krylov-Schur methods to calculate multiple eigenvalues of the λ-eigenvalue problem, of the Neutron Diffusion Equation (Bernal et al 2017a, Vidal-Ferrandiz et al 2014, Bernal et al 2018, Theler 2013, and Carreño et al 2017. Actually, one can find a great analysis of the application of Krylov-Schur methods for the calculation of different kinds of eigenvalue problems of the Neutron Diffusion Equation in Carreño et al 2017.…”

“…In this section, the author used the Nodal Collocation Method (Hébert 1987), with Legendre polynomials of order three. Actually, the author used a version of TRIVAC based on its version 5, which includes an eigensolver based on the SLEPc library (Bernal et al 2017a). The mesh used in TRIVAC is that of Figure 6.52.…”