Discontinuous Galerkin Methods for Convection-Dominated Problems

We present and analyze the Runge Kutta Discontinuous Galerkin method for numerically solving nonlinear hyperbolic systems. The basic method is then extended to convection-dominated problems yielding the Local Discontinuous Galerkin method. These methods are particularly attractive since they achieve formal high-order 0accuracy, nonlinear stability, and high parallelizability while maintaining the ability to handle complicated geometries and capture the discontinuities or strong gradients of the exact solution without producing spurious oscillations. The discussed methods are readily applied to 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. As a final example, consideration is given to the application of the discontinuous Galerkin method to the Hamilton-Jacobi equations.