Galerkin method for asymmetrically-weighted Hermite approximations applied to the Vlasov-Poisson system
arXiv:2607.28409
The paper proposes a Galerkin spectral method using asymmetrically-weighted Hermite bases for the Vlasov‑Poisson system, providing L2 stability and comparable computational cost to existing Petrov‑Galerkin approaches.
Abstract
We investigate a numerical method for the Vlasov-Poisson (VP) system utilizing asymmetrically-weighted (AW) Hermite bases in velocity space, which is an hyperbolic system. In particular, we concentrate on spectral methods in velocity. For the Hermite spectral form of the VP system, we analyze the resaon that the form with AW Hermite bases can be instable. To obtain L2 stability properties, we consider a Galerkin method intead of the classical Petrov-Galerkin method, which naturally ensures stability with respect to the L2 norm. We also present an equivalent form of the method that maintains a computational cost modest compared to that of the Petrov-Galerkin method. Finally, we present numerical simulations based on the proposed Hermite spectral method, showcasing its stability.