Convergence of spectral discretizations of the Vlasov-Poisson system
This provides rigorous convergence guarantees for spectral methods applied to kinetic plasma simulations, which is important for computational plasma physics but represents an incremental theoretical extension of existing numerical analysis techniques.
The authors prove convergence of a spectral discretization for the Vlasov-Poisson system, showing spectral error estimates under regularity assumptions. The method uses Hermite or Legendre polynomials in velocity and Fourier expansions in space, with a penalty term for boundary conditions.
We prove the convergence of a spectral discretization of the Vlasov-Poisson system. The velocity term of the Vlasov equation is discretized using either Hermite functions on the infinite domain or Legendre polynomials on a bounded domain. The spatial term of the Vlasov and Poisson equations is discretized using periodic Fourier expansions. Boundary conditions are treated in weak form through a penalty type term, that can be applied also in the Hermite case. As a matter of fact, stability properties of the approximated scheme descend from this added term. The convergence analysis is carried out in details for the 1D-1V case, but results can be generalized to multidimensional domains, obtained as Cartesian product, in both space and velocity. The error estimates show the spectral convergence, under suitable regularity assumptions on the exact solution.