Simulations of the Vlasov-Poisson system and the study of recurrence for the discontinuous Galerkin method

We describe the Runge-Kutta discontinuous Galerkin (RKDG) scheme\footnote{R. E.~Heath, I.~M.~Gamba, P.~J.~Morrison, and C.~Michler, J.\ Comp.\ Phys.\ {\bf 231}, 1140 (2012).} for the Vlasov-Poisson system that models collisionless plasmas. One-dimensional systems are emphasized. This numerical method used is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and be used with a positivity-preserving limiter that guarantees positivity of the distribution function. We compute the solutions using a high-order discontinuous Galerkin method for the Vlasov equation, and the classical representation by Green's function for the Poisson equation in the one-dimensional setting. We performed Fourier analysis to study recurrence of the discontinuous Galerkin methods on Cartesian meshes. Results from several benchmark test problems, such as Landau damping, two-stream instability and the KEEN (Kinetic Electrostatic Electron Nonlinear) wave, are given and interpreted.