An anisotropic nonlinear stabilization for finite element approximation of Vlasov-Poisson equations

We introduce a high-order finite element method for approximating the Vlasov-Poisson equations. This approach employs continuous Lagrange polynomials in space and explicit Runge-Kutta schemes for time discretization. To stabilize the numerical oscillations inherent in the scheme, a new anisotropic nonlinear artificial viscosity method is introduced. Numerical results demonstrate that this method achieves optimal convergence order with respect to both the polynomial space and time integration. The method is validated using classic benchmark problems for the Vlasov-Poisson equations, including Landau damping, two-stream instability, and bump-on-tail instability in a two-dimensional phase space.