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Alternating direction implicit type preconditioners for the steady state inhomogeneous Vlasov equation

Published online by Cambridge University Press:  20 February 2017

Markus Gasteiger*
Affiliation:
University of Innsbruck, Department of Mathematics, Technikerstr. 13, A-6020 Innsbruck, Austria
Lukas Einkemmer
Affiliation:
University of Innsbruck, Department of Mathematics, Technikerstr. 13, A-6020 Innsbruck, Austria
Alexander Ostermann
Affiliation:
University of Innsbruck, Department of Mathematics, Technikerstr. 13, A-6020 Innsbruck, Austria
David Tskhakaya
Affiliation:
Technical University of Vienna, Institute of Applied Physics, Fusion@ÖAW, Wiedner Hauptstr. 8-10/E134, A-1040 Wien, Austria
*
Email address for correspondence: markus.gasteiger@uibk.ac.at

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 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.

Type
Research Article
Copyright
© Cambridge University Press 2017 

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