Multilevel matrix-free preconditioner to solve linear systems associated with a the time-dependent SPN equations

Abstract

[EN] The evolution of the neutronic power inside of a nuclear reactor core can be approximated by means of the diffusive time-dependent simplified spherical harmonics equations (SPN). For the spatial discretization of these equations, a continuous Galerkin high order finite element method is applied to obtain a semi-discrete system of equations that is usually stiff. A semi-implicit time scheme is used for the time discretization and many linear systems are needed to be solved and previously, preconditioned. The aim of this work is to speed up the convergence of the linear systems solver with a multilevel preconditioner that uses different degrees of the polynomials used in the finite element method. Furthermore, as the matrices that appear in this type of system are very large and sparse, a matrix-free implementation of the preconditioner is developed to avoid the full assembly of the matrices. A benchmark transient tests this methodology. Numerical results show, in comparison with the block Gauss-Seidel preconditioner, an improvement in terms of number of iterations and the necessity of computational resources.