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We discuss a spectral method for the numerical solution of the Vlasov-Poissonsystem where the velocity space is decomposed by means of an Hermite basis. Wedescribe a semi-implicit time discretization that extends the range ofnumerical stability relative to an explicit scheme. We also introduce anddiscuss the effects of an artificial collisional operator, which is necessaryto take care of the velocity space filamentation problem, unavoidable incollisionless plasmas. The computational efficiency and the cost-effectivenessof this method are compared to a Particle-in-Cell (PIC) method in the case of atwo-dimensional phase space. The following examples are discussed: Langmuirwave, Landau damping, ion-acoustic wave, two-stream instability, and plasmaecho. The Hermite spectral method can achieve solutions that are several ordersof magnitude more accurate (at a fraction of the cost) with respect to the PICmethod.

On the velocity space discretization for the Vlasov-Poisson system:  comparison between Hermite spectral and Particle-in-Cell methods. Part 1:  semi-implicit scheme