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 some standard benchmark problems including the two-stream instability and the Landau damping test cases.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.
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