A study into discontinuous Galerkin methods for the second order wave equation

Abstract

There are numerous numerical methods for solving different types of partial differential equations (PDEs) that describe the physical dynamics of the world. For instance, PDEs are used to understand fluid flow for aerodynamics, wave dynamics for seismic exploration, and orbital mechanics. The goal of these numerical methods is to approximate the solution to a continuous PDE with an accurate discrete representation. The focus of this thesis is to explore a new Discontinuous Galerkin (DG) method for approximating the second order wave equation in complex geometries with curved elements. We begin by briefly highlighting some of the numerical methods used to solve PDEs and discuss the necessary concepts to understand DG methods. These concepts are used to develop a one- and two-dimensional DG method with an upwind flux, boundary conditions, and curved elements. We demonstrate convergence numerically and prove discrete stability of the method through an energy analysis.