Numerical Solutions for Partial Differential Equations of Conservation Laws

Abstract

The goal of this project, is to come up with a su cient background so thatwe can approach the current literature of research possessing the necessary tools and detailed understanding. We will concern ourselves with analyzing the discontinuous galerkin method (DGM) by looking at its background and formulation. We will deal with the theory of mathematical outlook of these equations rst and then solutions. This will cause us to emphasize more on the tools of mathematics that are very important in the of development, analyzing and successful utilization of the nite di erence method for the non-linear systems of conservation laws, in particular for problems involving Shallow Water Equations. The derivation of these equations will be provided. Also the shallow water equations will be given in both conservative and non-conservative form. The main type of method used in the approximation of di erential equations of this kind will be given i.e the nite difference method. We will later formulate the solutions to the shallow water equation in MATLAB.