Implicit Discontinuous Galerkin Method for Hyperbolic Equations

In this work a technique is presented for the numerical approximation of conservation laws. It is based both on the Runge-Kutta Discontinuous Galerkin method [3] and the Streamline Upwind Petrov-Galerkin method [2]. The proposed numerical scheme uses a discontinuous piecewise polynomial approximation in space and implicit backward Euler time stepping. Numerical oscillations within the discontinuous elements are controlled by adding a streamline diffusive term. An optimal relation between the time step (in terms of the CFL condition) and the size of the diffusion coefficient is analysed for numerical precision. The scheme is implemented using the object oriented programming philosophy based on the environment described in [4]. Accuracy and shock capturing abilities of the method are analysed in terms of two bidimensional model problems, the rotating hill problem and the backward facing step problem for the Euler equations of gas dynamics.