A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System based on Fourier, Legendre and Hermite Polynomials
For computational plasma physicists, this provides a spectral method with improved velocity discretization options, but the results are incremental as they confirm known parameter sensitivity without offering a solution.
This work extends a semi-Lagrangian spectral method for the Vlasov-Poisson system to use Legendre polynomials and Hermite functions in velocity, achieving second-order accuracy and good conservation properties on the two-stream instability benchmark. The Hermite case shows that a scaling parameter in the Gaussian weight significantly impacts accuracy, motivating adaptive strategies.
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.