A stable and efficient Galerkin spectral method with asymmetrically-weighted Hermite functions for the Vlasov-Poisson system

Abstract

We analyze a Galerkin spectral method applied to the Vlasov-Poisson (VP) system based on time-independent asymmetrically-weighted (AW) Hermite functions in velocity. The VP system is written as an hyperbolic system using AW Hermite functions in velocity. Unlike the classical Petrov-Galerkin spectral method, which is not stable, the proposed Galerkin spectral method admits a natural stability in the unweighted L2 norm. This stability property enables a rigorous convergence analysis of the method. For sufficiently regular solutions with exponential decay in velocity, we establish error estimates between the exact and numerical solutions and prove convergence of the Galerkin spectral method. In particular, the method achieves spectral convergence in Sobolev spaces, with convergence rates determined by the regularity of the exact solution. Since the Gram matrix arising from the proposed Galerkin method is dense, we derive an equivalent form that retains the same approximation properties while preserving the sparsity structure similar to the classical Petrov-Galerkin method.