Date of Award

2010

Degree Type

Report

Degree Name

Master of Science (MS)

Department

Mathematics and Statistics

First Advisor

Joseph Koebbe

Abstract

Plasmas are frequently treated as a single conducting fluid and modeled using the equations of magnetohydrodynamics. However, this regime works better for low-frequency plasmas. High-frequency plasmas may be modeled using the principles of kinetic theory. For plasmas with frequencies between these two extremes, a two-fluid approach can yield better results. In 2006, Ammar Hakim mathematically modeled a plasma with a set of equations called the five-moment ideal two-fluid equations. An attempt is made reproduce those results. A derivation of this set of equations by taking moments of the Boltzmann equation is presented. Electric and magnetic fields contribute to the source terms, so Maxwell's equations are coupled to the system. Finally, it is shown that single-fluid results can be obtained by taking suitable limits of the five-moment ideal two-fluid equations.

Assuming a fully ionized plasma of electrons and ions produces a hyperbolic system of sixteen equations. This hyperbolic system is approximately solved for a one-dimensional problem using a finite volume approach. Differing values in adjacent cells are treated as initial data for a Riemann problem and the jump is decomposed into waves. An approximate Riemann solver created through Roe averaging is used to numerically construct these waves. Source terms are solved for by a Strang splitting. The divergence constraints found in Maxwell's equations can be handled in two different ways. The first approach, which has not been studied by the author, is to use a different set of Maxwell's equations called the perfectly hyperbolic Maxwell's equations (PHM). This approach introduces correction potentials to ensure that the divergence constraints are satisfied throughout the numerical simulation. The second approach is to choose specific values of the charge to mass ratio for the ions and electrons.

The software CLAWPACK is used for all simulations. A simple shock tube problem with one species and no source terms is first solved using sample code to illustrate some behavior of the larger system. Finally, an attempt is made at solving the same shock tube problem with two species and source terms. The resulting time ODE from the Strang splitting is solved using methods of various accuracy. It is shown that the time step required for stability is too small to be practical and thus a different approach to the problem, such as homogenized wavelet refinement, is needed.

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