Convergence of spectral discretizations of the Vlasov-Poisson system

TL;DR

This paper proves the spectral convergence of a discretization scheme for the Vlasov-Poisson system, using Hermite or Legendre functions for velocity and Fourier expansions for space, with a detailed 1D-1V analysis.

Contribution

It introduces a convergence proof for spectral discretizations of the Vlasov-Poisson system with boundary treatment via penalty terms, applicable to multidimensional domains.

Findings

Spectral convergence under regularity assumptions

Stability derived from penalty boundary treatment

Method applicable to multidimensional problems

Abstract

We prove the convergence of a spectral discretization of the Vlasov-Poisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain. The spatial term of the Vlasov and Poisson equations is discretized using periodic Fourier expansions. Boundary conditions are treated in weak form through a penalty type term, that can be applied also in the Hermite case. As a matter of fact, stability properties of the approximated scheme descend from this added term. The convergence analysis is carried out in details for the 1D-1V case, but results can be generalized to multidimensional domains, obtained as Cartesian product, in both space and velocity. The error estimates show the spectral convergence, under suitable regularity assumptions on the exact solution.