Off-campus South Dakota State University users: To download campus access theses, please use the following link to log into our proxy server with your South Dakota State University ID and password.

Non-South Dakota State University users: Please talk to your librarian about requesting this thesis through interlibrary loan.

Document Type

Dissertation - University Access Only

Award Date


Degree Name

Doctor of Philosophy (PhD)


Mathematics and Statistics

First Advisor

Matthew Biesecker


Previous models for generating synthetic thunder lacked wave interaction due to lightning channel tortuosity. In this work, a two-dimensional CFD-type model based on the Navier-Stokes equations which includes the effects of shear viscosity, bulk viscosity, thermal conductivity, wind shear, refraction, and is capable of applying the effect of dispersion clue to molecular relaxation of nitrogen and oxygen, is selected for utilization. The model is a set of six coupled equations, along with two equations of state to close the system. The model is augmented to be numerically solved using a hybrid scheme in space as well as the technique of adaptive mesh refinement (AMR). The hybrid scheme is based on the fusion of a fourth-order accurate dispersion-relation preserving (DRP) scheme with a weighted essentially non-oscillatory (WE O) scheme. Third-order and fifth-order accurate WENO schemes are explored for service. The WENO scheme requires more computations than the DRP scheme but it is capable of propagating shock waves stably. The DR P scheme exhibits oscillations at discontinuities, but its computations are less complex than the WENO scheme. Therefore, the hybrid scheme must calculate a gradient of the domain and choose to use the WENO scheme where discontinuous waves may exist and the DRP scheme where continuous waves or no waves exist. A Runge-Kutta scheme is used to march the model through time. AMR is implemented through the Structured Adaptive Mesh Refinement Application Infrastructure (SAMRAI) developed by Lawrence Livermore National Laboratory. AMR allows the computer to focus computational effort on areas of the domain which will be the most beneficial. The result of this is the capability of the computer to handle a larger domain size than it could with a static mesh while maintaining comparable accuracy. To insert the complex geometry of a lightning channel source a combination of mathematical functions is developed to efficiently insert the source and allow it to be sufficiently smooth to promote stability. The final result is a model that is capable of running on a large-scale parallel platform that can reproduce the acoustic signature of arbitrary tortuous lightning channel geometries including the effects of wave interaction over adequately-sized domains. However, lack of computational resources reduce the ability to achieve truly phenomenologically consistent results using the model.

Library of Congress Subject Headings

Thunder -- Mathematical models
Acoustic phenomena in nature



Number of Pages



South Dakota State University


Copyright © 2012 Jonathan S. Rood