  We describe a spectral method for the numerical solution of the
Vlasov-Poisson system where the velocity space is decomposed by means of an
Hermite basis, and the configuration space is discretized via a Fourier
decomposition. The novelty of our approach is an implicit time discretization
that allows exact conservation of charge, momentum and energy. The
computational efficiency and the cost-effectiveness of this method are compared
to the fully-implicit PIC method recently introduced by Markidis and Lapenta
(2011) and Chen et al. (2011). The following examples are discussed: Langmuir
wave, Landau damping, ion-acoustic wave, two-stream instability. The
Fourier-Hermite spectral method can achieve solutions that are several orders
of magnitude more accurate at a fraction of the cost with respect to PIC. This
paper concludes the study presented in Camporeale et al. (2013) where the same
method has been described for a semi-implicit time discretization, and was
compared against an explicit PIC.
