An introduction to the Discontinuous Galerkin method for convection-dominated problems

Abstract

In these notes, we study the Runge Kutta Discontinuous Galerkin method for numericaly solving nonlinear hyperbolic systems and its extension for convection-dominated problems, the so-called Local Discontinuous Galerkin method. Examples of problems to which these methods can be applied are the Euler equations of gas dynamics, the shallow water equations, the equations of magneto-hydrodynamics, the compressible Navier-Stokes equations with high Reynolds numbers, and the equations of the hydrodynamic model for semiconductor device simulation. The main features that make the methods under consideration attractive are their formal high-order accuracy, their nonlinear stability, their high parallelizability, their ability to handle complicated geometries, and their ability to capture the discontinuities or strong gradients of the exact solution without producing spurious oscillations. The purpose of these notes is to provide a short introduction to the devising and analysis of these discontinuous Galerkin methods.