Arbitrary-Order Time-Accurate Semi-Lagrangian Spectral Approximations of the Vlasov-Poisson System

The Vlasov-Poisson system, modeling the evolution of non-collisional plasmas in the electrostatic limit, is approximated 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 truncated Taylor series that can be, in principle, of any order of accuracy, or by coupling the phase space discretization with high-order accurate Backward Dierentiation 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, and the resulting method has excellent conservation properties. Indeed, we can prove that the total number of particles, which is proportional to the total mass and charge, is conserved up to the machine precision. Series of numerical experiments are performed in order to assess the real performance. In particular, comparisons with standard benchmarks are examined.