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