Ringing instabilities in particle methods

Particle methods have been used in fields ranging from fluid dynamics to plasma physics. The Particle-In-Cell method and the family of methods that are an extension of it are a combination of both Lagrangian and Eularian methods. In this thesis, we present a brief survey of some of the methods and their key components. We show the different methods by which spatial derviates are computed. We propose a method of showing how the so-called "ringing instabilies" associated with particle methods arise and a means to remove them. We also propose that the underlying nodal scheme plays a key role in the stability of the method. Lastly, dierent particle methods are explored through numerical simulations and compared against an analytic solution.