A Semi-Lagrangian Spectral Method for the Vlasov–Poisson System Based on Fourier, Legendre and Hermite Polynomials

Abstract

In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF time-stepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on the standard two-stream instability benchmark problem. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.

On the Use of Hermite Functions for the Vlasov–Poisson System

“…|â (k) 1 | in (16) considerations, and the slope of the "segment" starting at T = 15 agrees with the expectancy [1,Chapter 5]. As done in [9], we perform a series of experiments using less degrees of freedom than those actually necessary to resolve accurately the equation. In practice, we set N = M = 2 4 .…”

“…The approximation scheme can be easily adjusted by modifying nodes and weights of the Gaussian formula, through a multiplication by suitable constants. The difficulty in the implementation is practically the same, but, as observed in [9], the results are quite sensitive to the variation of α.…”

“…1. The purpose in [9] was to check what happens by varying the parameter α in the Hermite weight exp (−α 2 v 2 ). The conclusions are that the approximate solution is very sensitive to the choice of α and that there are values of α that perform better than others.…”

“…The approximation introduced in [8] has been initially developed and tested on Fourier-Fourier periodic discretizations, for both variables in the phase space. In the successive paper [9], the approximation in the variable v has been approached with the help of Hermite functions, i.e., Hermite polynomials multiplied by the Gaussian weight exp (−v 2 ).…”

A conservative semi-Lagrangian finite difference WENO scheme based on exponential integrator for one-dimensional scalar nonlinear hyperbolic equations

“…The classical backward SL scheme traces the characteristics back to a previous time level and then updates the solution with interpolation, or polynomial reconstruction. Based on its solution space, there are SL FD schemes [26,29], SL finite volume (FV) schemes [9,12], SL spectral element methods [32,13], SL discontinuous Galerkin (DG) finite element methods [23,3], SL particle methods [8], etc. For SL schemes, mass conservation is an important property for certain applications, such as in kinetic simulations [9,2], weather forecasting [27,15], fluid dynamics [29,12], etc.…”