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International Journal of Differential Equations
Volume 2014, Article ID 178625, 8 pages
http://dx.doi.org/10.1155/2014/178625
Research Article

Multiscale Splitting Method for the Boltzmann-Poisson Equation: Application to the Dynamics of Electrons

Department of Physics, Ernst-Moritz-Arntz University of Greifswald, Domstraße 14, 17487 Greifswald, Germany

Received 15 August 2013; Revised 20 January 2014; Accepted 21 January 2014; Published 3 March 2014

Academic Editor: Shuyu Sun

Copyright © 2014 Jürgen Geiser. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present a model based on dynamics of electrons in a plasma using a simplified Boltzmann equation coupled with Poisson’s equation. The motivation arose from simulating active plasma resonance spectroscopy, which is used for plasma diagnostic techniques; see Braithwaite and Franklin (2009), Lapke et al. (2010), and Oberrath et al. (2011). Mathematically, we are interested in designing splitting methods for the model problem. While the full Boltzmann equation is delicate to solve, we decouple it into a transport and collision part, which are then solved in different ways. First we reduce it to a simplified transport-collision equation and start to analyse the abstract Cauchy problem using semigroup methods. Second, we pass to the coupled transport and collision model and apply the splitting ideas, resecting the different discretization schemes. The results are discussed first with numerical experiments and then we verify the underlying theoretical novelties.