A novel, structure-preserving, second-order-in-time relaxation scheme for Schrödinger-Poisson systems

We introduce a new structure preserving, second order in time relaxation-type scheme for approximating solutions of the Schr\"odinger-Poisson system. More specifically, we use Crank-Nicolson as stepping mechanism, whilst nonlinearity is handled by means relaxation approach spirit \cite{Besse, KK} nonlinear Schr\"odinger equation. For spatial discretisation standard conforming finite element scheme. The resulting explicit with respect to nonlinearity, i.e. it requires solution linear system each time-step, and satisfies discrete versions system's mass conservation energy balance laws constant meshes. seen be time. conclude presenting some numerical experiments, including an example from cosmology variable time-steps which demonstrate effectiveness robustness