Numerical solution of the Vlasov–Poisson system using generalized Hermite functions

Two different spectral approaches for solving the nonlinear Vlasov–Poisson equations are presented and discussed. The first approach is based on a standard spectral Galerkin method (SGM) using Hermite functions in the velocity space. The second method which belongs to the family of pseudospectral methods (SCM) uses Gauss–Hermite collocation points for the velocity discretization. The high-dimensional feature of these equations and the suspected presence of small scales in the solution suggested us to employ these methods that provide high order accuracy while considering a “small” number of ad hoc basis functions. The scaled Hermite functions allow us to treat the case of unbounded domains and to properly recover Gaussian-type distributions. Some numerical results on usual test cases are shown and prove the good agreement with the theory.