# Variational Integrators for the Vlasov-Poisson System

Recently, an extension of the variational integrator framework has been proposed, which allows the application of the method even to nonvariational partial differential equations. In this work, this new method is applied to the Vlasov-Poisson system in Eulerian coordinates, in order to obtain grid-based numerical schemes, which respect important conservation laws on the discrete level. More precisely, the resulting integrators preserve mass, momentum, energy and the $$L^{2}$$ norm of the distribution function exactly. Due to the absence of dissipation, small scale structures appear in the solution, which eventually cannot be resolved and render the simulation unstable. To mitigate this problem, we apply a collision operator, which adds a minimum amount of diffusion in velocity space, thereby dissipating the $$L^{2}$$ norm of the distribution function, but retaining exact conservation of mass, momentum and energy. As most of the variational integrators obtained in this work are implicit, suitable preconditioning strategies are needed in order to solve the discrete Vlasov-Poisson system efficiently. We devise a new preconditioner based on a tensor product decomposition of the discrete Poisson bracket in the Vlasov equation, which is very effective so that convergence of the linear solver is achieve after only few iterations. Several numerical examples corroborate the favourable properties of the variational integrators.