A new Besse-type relaxation scheme for the numerical approximation of the Schrödinger-Poisson system

We introduce a new second order in time Besse-type relaxation scheme for approximating solutions of the Schr\"odinger-Poisson system. More specifically, we use the Crank-Nicolson scheme as a time stepping mechanism, the standard conforming finite element method for the spatial discretization whilst the nonlinearity is handled by means of a relaxation approach similar to the one introduced by Besse for the nonlinear Schr\"odinger equation \cite{Besse}. We prove that discrete versions of the system's conservation laws hold and we conclude by presenting some numerical experiments, including an example from cosmology, that demonstrate the effectiveness and robustness of the new scheme.