Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov-Poisson system

Abstract

The Vlasov-Poisson system, modeling the evolution of non-collisional plasmas in the electrostatic limit, is approx- imated by a Semi-Lagrangian technique. Spectral methods of periodic type are implemented through a collocation approach. Groups of particles are represented by the Fourier Lagrangian basis and evolve, for a single timestep, along an high-order accurate representation of the local characteristic lines. The time-advancing technique is based on Taylor developments that can be, in principle, of any order of accuracy, or by coupling the phase space discretiza- tion with high-order accurate Backward Differentiation Formulas (BDF) as in the method-of-lines framework. At each timestep, particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs. Thus, these techniques combine excellent features of spectral approximations with high-order time integration. Series of numerical experiments are performed in order to assess the real performance. In particular, comparisons with standard benchmarks are examined.

25 pages, 8 figures

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• The multi-dimensional Hermite-discontinuous Galerkin method for the Vlasov-Maxwell equations
• On the stability of conservative discontinuous Galerkin/Hermite spectral methods for the Vlasov-Poisson system
• 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
• Splitting methods for Fourier spectral discretizations of the strongly magnetized Vlasov-Poisson and the Vlasov-Maxwell system