Convergence of Galerkin approximations based on asymmetrically-weighted Hermite functions for the Vlasov-Poisson system

The convergence of Galerkin approximations based on asymmetrically-weighted (AW) Hermite functions, applied to the Vlasov-Poisson (VP) system, is analyzed in this work. The VP system is written as an hyperbolic system using Hermite functions in velocity. To obtain stability properties of the numerical methods, we consider an unweighted inner product, and establish naturely the stability with respect to the unweighted norm. Consequently, we prove the convergence of the method. Furthermore, we derive error estimates between the numerical solution and the smooth solution of the VP system. Our results confirm that the Galerkin approximations based on AW Hermite functions exhibit spectral accuracy in Sobolev spaces.