Doctoral Dissertations
Date of Award
8-2025
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Major
Mathematics
Major Professor
Cory D. Hauck
Committee Members
Steven M. Wise, Abner Salgado, Ryan Glasby
Abstract
This dissertation develops numerical methods for kinetic equations modeling gases and plasmas, with a focus on general multi-species particle mixtures, and is divided into two main parts.
Part I presents numerical strategies for Bhatnagar--Gross--Krook (BGK) models, which approximate the Boltzmann equation using a nonlinear, nonlocal relaxation operator. The methods are designed to handle numerical stiffness near the hydrodynamic limit, where the Knudsen number is small and collisions are frequent, and to capture features like discontinuities and shocks. A high-order finite volume framework for phase space flux reconstruction is combined with implicit-explicit Runge--Kutta time-stepping methods to construct a robust solver for the single-species model.
Part II addresses multi-species kinetic models in which collision frequencies depend on species temperatures. This dependence introduces significant complexity into the implicit update of collision terms. A system of moment equations governing the evolution of bulk velocities, energies, and temperatures of a multi-species BGK (M-BGK) model is analyzed in the spatially homogeneous setting. Global existence and uniqueness results are established by proving a uniform-in-time lower bound on species temperatures, which also provides estimates of convergence rates toward equilibrium.
To numerically advance the stiff moment system of the M-BGK model, a nonlinear fixed-point iterative method, motivated by the Gauss--Seidel method, is introduced and shown to converge under mild time step constraints, independent of the system’s stiffness. The iterative method enables efficient implicit updates within the semi-implicit framework.
Finally, the dissertation introduces a multi-species Lenard--Bernstein (M-LB) model, which approximates the Fokker--Planck collision operator for plasmas. The model includes tunable parameters that allow matching of momentum and temperature relaxation rates derived from the Boltzmann operator with Coulomb interactions. Under natural physical assumptions, the model satisfies conservation laws, dissipates entropy, and obeys an H-Theorem. A similar iterative strategy is applied to update the M-LB model, enabling simulations of relaxation toward equilibrium.
Overall, this dissertation provides a unified framework for the analysis and numerical simulation of kinetic models with temperature-dependent interactions, with implications for modeling realistic gas and plasma mixtures across a range of physical regimes.
Recommended Citation
Habbershaw, Evan, "Numerical Methods for Multi-Species Kinetic Models with Approximate Collision Operators. " PhD diss., University of Tennessee, 2025.
https://trace.tennessee.edu/utk_graddiss/12714
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