Kinetic simulations of relativistic vlasov systems
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Authors
Gorman, Jennifer, Kathleen
Issue Date
2024-08
Type
Electronic thesis
Thesis
Thesis
Language
en_US
Keywords
Mathematics
Alternative Title
Abstract
This thesis presents high-order discretization techniques for the relativistic continuum kineticVlasov equation in up to 6D phase space. These algorithms have been implemented in the
highly parallel code LOKI. The details of its discretization and implementation are described,
and vigorous verification is carried out using the method of manufactured solutions. The
topic then moves to Landau damping and the dispersion relation of Langmuir waves. A new
technique is introduced to evaluate and solve the dispersion relation which allows for any
equilibrium function and the inclusion of relativistic effects. These results are then compared
to results from LOKI simulations. LOKI is a highly parallel code used to approximate the continuum kinetic Vlasovequation and the associated electrostatic Vlasov-Poisson system and electromagnetic Vlasov-Maxwell system. It is made high order accurate by utilizing up to sixth-order accurate
explicit Runge Kutta (RK) schemes for time integration, and the minimally dissipative
upwinding scheme BWENO to approximate the phase space derivatives. In previous literature, the details of LOKI’s initial implementation in 2+2D (two spatial and two velocity
dimensions) without dissipation and no Maxwell were outlined. In the course of this thesis,
LOKI has been extended to include a variety of dimensional configurations, up to 3+3D,
and the ability to simulate relativistic effects were included. The implementation of such in
3+3D with sixth order accuracy is described in detail for both Vlasov-Poisson and Vlasov-Maxwell. Each physical system is then verified thoroughly for all dimensional configurations
(1+1D, 1+2D, 1+3D, 2+2D, 2+3D, and 3+3D) with and without relativistic effects using
the method of manufactured solutions. These checks show the code is achieving both fourth- and sixth-order convergence in numerical errors for all configurations. Although manufactured solutions are useful for catching coding errors, verification ofnumerical models using theoretical results is required for thorough confidence in simulation
accuracy. An analytical method which evaluates the Langmuir wave dispersion relation to
machine precision is introduced, which suggests integrating directly over the Landau contour, with the inclusion of a residue around the pole in the complex plane when necessary.
This method allows for more general equilibrium functions, and the easy introduction of
relativistic effects into the formulation. The standard one dimensional non-relativistic dispersion relation found with this new method using a Maxwell-Boltzmann distribution function is compared to the exact solution, as well as previous analytical solutions for the same,
and shows excellent agreement to machine precision (approximately 16 digits of accuracy).
Results are then computed in 1D, 2D, and 3D for both the relativistic and non-relativistic
Maxwell-Boltzmann equilibrium function, as well as for the relativistic Maxwell-Juttner equilibrium, and verified through comparison with Landau damping results of the electric field
of kinetic simulations from LOKI. Using these analytical results, observations are made of
Landau damping in highly relativistic plasmas with the Vlasov-Poisson model.
Description
August 2024
School of Science
School of Science
Full Citation
Publisher
Rensselaer Polytechnic Institute, Troy, NY