ADI type preconditioners for the steady state inhomogeneous Vlasov equation

Abstract

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.