Discontinous Galerkin Methods for Conservation Laws, - with and without fractional diffusion

This thesis was submitted on July 5'th 2018 as the Master's thesis for Alexander N Sigurdsson in Industrial Mathematics at the Department of Mathematical Sciences at The Norwegian University of Science and Technology (NTNU). In this thesis we apply the Discontinuous Galerkin (DG) methods on scalar conservation laws with and without fractional diffusion. The supporting theory of Discontinuous Galerkin methods is taken from \cite{Cockburn1999} and \cite{Hesthaven2008} while the discretizing of the fractional Laplacian is shown in \cite{Jakobsen}. More general theory on numerical solutions of PDE's is found from \cite{Quarteroni2014} and \cite{leveque}. The first part of the thesis gives an introduction to DG methods for scalar conservation laws. We then employ the methods to solve specific conservation laws with both linear and non-linear flux. Here we also introduce an explicit numerical time integration scheme, a Runge-Kutta method with TVD (Total Variation Diminishing) properties (RKTVD method) and show how to use slope-limiting techniques to avoid spurious oscillations that occur for higher-order methods. Numerical examples and results are shown for both linear and non-linear flux up to 2'nd order. The second part tackles the addition of a fractional diffusion operator specifically the fractional Laplacian. By using a result where the fractional Laplacian can be written as a singular integral we are able to discretize it and find solutions of the fractional conservation law. Numerical examples with fractional diffusion are shown for both linear and non-linear flux up to 1'st order. The appendix gives a more thorough walk-through of how the integrals are analytically calculated for the fractional Laplacian, and how one can calculate these integrals numerically