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

Inside a nuclear reactor core, the neutronic power distribution can be approximated by means of the multigroup time-dependent  simplified spherical harmonics equations. In particular, this work uses a formulation where the time derivatives of the even spherical harmonics moments are assumed equal to zero. This treatment yields to diffusive equations of order two that only depend on the position and time.For the spatial discretization of the equations, a continuous Galerkin high order finite element method is applied. In the time discretization, two sets of equations appear: one related to the neutron moments and the other related to the delayed neutron precursor concentrations. Moreover, these time differential equations are usually stiff. Thus, a semi-implicit time scheme must be proposed that needs to solve several linear systems in each time-step. And generally, these systems must be preconditioned.The main aim of this work is to speed up the convergence of the linear systems solver with a multilevel preconditioner that uses different levels of energy, spherical moments and degrees in the finite element method. Furthermore, the matrices that appear in this type of system are large and sparse. A matrix-free implementation is used to avoid the full assembly of the matrices. Therefore, the multilevel preconditioner must be applied by matrix-vector products.Different benchmark transients test these techniques. Numerical results show, in the comparison with classical methodologies, an improvement in terms of memory requested and time needed to obtain the solution.