A Numerical Analysis of Smoothed Particle Hydrodynamics

Abstract

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.