  Spectral approximation based on Hermite-Fourier expansion of the
Vlasov-Poisson model for a collisionless plasma in the electro-static limit is
provided, by including high-order artificial collision operators of
Lenard-Bernstein type. These differential operators are suitably designed in
order to preserve the physically-meaningful invariants (number of particles,
momentum, energy). In view of time-discretization, stability results in
appropriate norms are presented. In this study, necessary conditions link the
magnitude of the artificial collision term, the number of spectral modes of the
discretization, as well as the time-step. The analysis, carried out in full for
the Hermite discretization of a simple linear problem in one-dimension, is then
partly extended to cover the complete nonlinear Vlasov-Poisson model.
