Strategies of Preconditioner Updates for Sequences of Linear Systems Associated with the Neutron Diffusion

The time-dependent neutron diffusion equation approximates the neutronic power evolution inside a nuclear reactor core. Applying a Galerkin finite element method for the spatial discretization of these equations leads to a stiff semi-discrete system of ordinary differential equations. For time discretization, an implicit scheme is used, which implies solving a large and sparse linear system of equations for each time step. The GMRES method is used to solve these systems because of its fast convergence when a suitable preconditioner is provided. This work explores several matrix-free strategies based on different updated preconditioners, which are constructed by low-rank updates of a given initial preconditioner. They are two tuned preconditioners based on the bad and good Broyden’s methods, initially developed for nonlinear equations and optimization problems, and spectral preconditioners. The efficiency of the resulting preconditioners under study is closely related to the selection of the subspace used to construct the update. Our numerical results show the effectiveness of these methodologies in terms of CPU time and storage for different nuclear benchmark transients, even if the initial preconditioner is not good enough.