  A discontinuous Galerkin method for approximating the Vlasov-Poisson system
of equations describing the time evolution of a collisionless plasma is
proposed. The method is mass conservative and, in the case that piecewise
constant functions are used as a basis, the method preserves the positivity of
the electron distribution function and weakly enforces continuity of the
electric field through mesh interfaces and boundary conditions. The performance
of the method is investigated by computing several examples and error estimates
associated system's approximation are stated. In particular, computed results
are benchmarked against established theoretical results for linear advection
and the phenomenon of linear Landau damping for both the Maxwell and Lorentz
distributions. Moreover, two nonlinear problems are considered: nonlinear
Landau damping and a version of the two-stream instability are computed. For
the latter, fine scale details of the resulting long-time BGK-like state are
presented. Conservation laws are examined and various comparisons to theory are
made. The results obtained demonstrate that the discontinuous Galerkin method
is a viable option for integrating the Vlasov-Poisson system.
