Spectral Methods for Partial Differential Equations that Model Shallow Water Wave Phenomena

Abstract

Mathematical models for waves on shallow water surfaces has been of interest to researchers dating back to the 1800's. These models are governed by partial differential equations, and many of them have rich mathematical structure as well as real world applications. This thesis explores a class of numerical techniques for partial differential equations called spectral methods. One can use these spectral methods to approximate solutions to many partial differential equations that model wave type phenomena. Of particular interest are the KdV, BBM, Camassa-Holm, Boussinesq systems, Shallow Water, and Serre Green- Naghdi equations. For all examples presented Matlab code is provided. These files will be uploaded to the GitHub page https://github.com/msfabien/.