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M. Gasteiger, L. Einkemmer, A. Ostermann, D. Tskhakaya:
"Alternating direction implicit type preconditioners for the steady state inhomogeneous Vlasov equation";
Journal of Plasma Physics, 83 (2017), 7058301071 - 70583010713.

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
such as the generalized minimal residual method (GMRES) or relaxation schemes)
is computationally expensive. In the former case the slowest time scale 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 alternating direction implicit 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 speed-up of close to two orders of magnitude (even for intermediate grid sizes) with
respect to the unpreconditioned case. In addition, we discuss the characteristics of
these numerical methods and show the results for a number of numerical simulations.

fusion plasma, plasma simulation