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

TL;DR

This paper develops a semi-Lagrangian spectral method using Fourier, Legendre, and Hermite polynomials for the Vlasov-Poisson system, demonstrating good conservation and accuracy in numerical tests, with insights on parameter dependence.

Contribution

It extends spectral methods for the Vlasov-Poisson system by incorporating Legendre and Hermite functions, and analyzes their numerical behavior and conservation properties.

Findings

Method achieves second-order accuracy in time.

Good conservation properties observed in numerical tests.

Parameter choice in Hermite functions significantly affects accuracy.

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.