ADI type preconditioners for the steady state inhomogeneous Vlasov equation

摘要

The purpose of the current work is to find numerical solutions of the steadystate inhomogeneous Vlasov equation. This problem has a wide range ofapplications in the kinetic simulation of non-thermal plasmas. However, thedirect application of either time stepping schemes or iterative methods (suchas Krylov based methods like GMRES or relexation schemes) is computationallyexpensive. In the former case the slowest timescale in the system forces us toperform a long time integration while in the latter case a large number ofiterations is required. In this paper we propose a preconditioner based on anADI type splitting method. This preconditioner is then combined with both GMRESand 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 closeto two orders of magnitude (even for intermediate grid sizes) with respect tothe not preconditioned case. In addition, we discuss the characteristics ofthese numerical methods and show the results for a number of numericalsimulations.