David A. Fulk

Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Department of Mathematics and Statistics

First Advisor

Dennis W. Quinn, PhD


This dissertation studies the numerical method of Smoothed Particle Hydrodynamics SPH as a technique for solving systems of conservation equations. The research starts with a detailed consistency analysis of the method. Higher dimensions and non-smooth functions are considered in addition to the smooth one dimensional case. A stability analysis is then performed. Using a linear technique, an instability is found. Solutions are proposed to resolve the instability. Also a total variation stability analysis is performed leading to a monotone form of SPH. The concepts of consistency and stability are then used in a convergence proof. This proof uses lemmas derived from the Lax-Wendroff theorem in finite differences. The numerical analysis of the method is concluded with a study of the SPH kernel function. Measures of merit are derived for SPH kernels and these are used to show bell-shaped kernels to be superior over other shaped kernels. Three second-order time schemes are applied to SPH to provide a full discretization of the problem these are Lax-Wendroff, central, and Shu schemes. In addition a lower-order SPH Lax-Friedrichs type form is developed. This method is used in proposing the use of flux-limited hybrid methods in SPH to resolve shocks.

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The author's Vita page is omitted.