  We present the design and implementation of an L2-stable spectral method for
the discretization of the Vlasov- Poisson model of a collisionless plasma in
one space and velocity dimension. The velocity and space dependence of the
Vlasov equation are resolved through a truncated spectral expansion based on
Legendre and Fourier basis functions, respectively. The Poisson equation, which
is coupled to the Vlasov equation, is also resolved through a Fourier
expansion. The resulting system of ordinary differential equation is
discretized by the implicit second-order accurate Crank-Nicolson time
discretization. The non-linear dependence between the Vlasov and Poisson
equations is iteratively solved at any time cycle by a Jacobian-Free
Newton-Krylov method. In this work we analyze the structure of the main
conservation laws of the resulting Legendre-Fourier model, e.g., mass,
momentum, and energy, and prove that they are exactly satisfied in the
semi-discrete and discrete setting. The L2-stability of the method is ensured
by discretizing the boundary conditions of the distribution function at the
boundaries of the velocity domain by a suitable penalty term. The impact of the
penalty term on the conservation properties is investigated theoretically and
numerically. An implementation of the penalty term that does not affect the
conservation of mass, momentum and energy, is also proposed and studied. A
collisional term is introduced in the discrete model to control the
filamentation effect, but does not affect the conservation properties of the
system. Numerical results on a set of standard test problems illustrate the
performance of the method.
