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

Jürgen Geiser

doi:10.1155/2014/178625

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.

Highlights

Our motive arose from studying the simulation of active plasma resonance spectroscopy, a well-established plasma diagnostic technique
We present a model based on dynamics of electrons in a plasma using a simplified Boltzmann equation coupled with Poisson’s equation
While the full Boltzmann equation is delicate to solve, we decouple it into a transport and collision part, which are 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

Summary

Introduction

Our motive arose from studying the simulation of active plasma resonance spectroscopy, a well-established plasma diagnostic technique To study this technique with simulation models, we concentrate on an abstract kinetic model that describes the dynamics of the electrons in plasma by using a Boltzmann equation. While finite difference schemes are applied to the transport parts, the collision part is solved with numerical integration schemes. The underlying splitting scheme is theoretically discussed as an abstract Cauchy problem. We will discuss the description of a positive semigroup, which helps carry out the numerical estimates in the splitting schemes. A numerical method is discussed with respect to separate differential and integral parts of the equations. The numerical approximation of the abstract splitting scheme is made by applying an iterative splitting method of the second order.

Mathematical Model

Semigroups for Transport Equations

Splitting Schemes

Numerical Integration of the Integrodifferential Part

Conclusions and Discussion