  The purpose of the current work is to find numerical solutions of the steady
state inhomogeneous Vlasov equation. This problem has a wide range of
applications in the kinetic simulation of non-thermal plasmas. However, the
direct application of either time stepping schemes or iterative methods (such
as Krylov based methods like GMRES or relexation schemes) is computationally
expensive. In the former case the slowest timescale in the system forces us to
perform a long time integration while in the latter case a large number of
iterations is required. In this paper we propose a preconditioner based on an
ADI type splitting method. This preconditioner is then combined with both GMRES
and Richardson iteration. The resulting numerical schemes scale almost ideally
(i.e. the computational effort is proportional to the number of grid points).
Numerical simulations conducted show that this can result in a speedup of close
to two orders of magnitude (even for intermediate grid sizes) with respect to
the not preconditioned case. In addition, we discuss the characteristics of
these numerical methods and show the results for a number of numerical
simulations.
