Abstract
In this thesis, we delve into conservation properties of wave equations. In particular, the equation of interest is the generalized Korteweg-de Vries (KdV) equation. We develop, analyze, and implement a new conservative Discontinuous Galerkin (DG) method for reproducing the solitary wave solutions of the generalized KdV Equation. The exact solutions of all KdV equations conserve the mass, energy and Hamiltonian. Numerically, our method is able to conserve these three quantities. In fact, to our knowledge, it is the first DG method that is able to conserve these three quantities which are important for accurate simulation of the long time evolution of these waves. In order to accomplish this, our novel idea is to introduce two new unknowns into the system in order to enforce the conservation of energy and Hamiltonian. These unknowns are stabilization parameters in our numerical flux. Herein, we also prove the conservation properties of our scheme which are further confirmed by numerical testing. This idea of defining penalization parameters implicitly can serve as a framework for designing numerical methods for other problems that have properties in need of conservation for accurate simulation. Furthermore, we design a new conservative DG method for the Hirota-Satsuma (HS) coupled KdV system by following the framework of our conservative DG method for KdV equations. We show that we can apply the same methodology to achieve desirable conservation properties of the HS-KdV system.