An Implementation of the Relativistic Hydrodynamic Equations in Conservative Form Using DoGPack

Abstract

Let g_μν be the metric associated with a stationary spacetime. In the 3 + 1 splitting of spacetime, this allows us to cast the relativistic hydrodynamic equations as a balance law of the form q_,t + ∇· F ( q) = S , which is a system of hyperbolic partial differential equations. These hyperbolic equations admit shocks and rarefactions in their weak solutions. Because of this, we employ a Runge-Kutta Discontinuous Galerkin method in both Minkowski and Schwarzschild spacetimes through the use of the Discontinuous Galerkin Package. In this thesis, we give a quick background on topics in general relativity necessary to implement the method, as well as details on the DG method itself. We present tests of the method in the form of shock tube tests and smooth flow into a black hole to show its versatility.