  We construct the approximate solutions to the Vlasov--Poisson system in a
half-space, which arises in the study of the quasi-neutral limit problem in the
presence of a sharp boundary layer, referred as to the plasma sheath in the
context of plasma physics. The quasi-neutrality is an important characteristic
of plasmas and its scale is characterized by a small parameter, called the
Debye length.
  We present the approximate equations obtained by a formal expansion in the
parameter and study the properties of the approximate solutions.
  Moreover, we present numerical experiments demonstrating that the approximate
solutions converge to those of the Vlasov--Poisson system as the parameter goes
to zero.
