Convergence of spectral discretizations of the Vlasov-Poisson system

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.

References in corpus (2)

• Fourier-Hermite spectral representation for the Vlasov-Poisson system in the weakly collisional limit
• A Legendre-Fourier spectral method with exact conservation laws for the Vlasov-Poisson system

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• A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System based on Fourier, Legendre and Hermite Polynomials
• Physics-based adaptivity of a spectral method for the Vlasov-Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space
• Conservative polynomial approximations and applications to Fokker-Planck equations