A Hybridized Discontinuous Galerkin Method for Unsteady Flows with Shock-Capturing

We present a hybridized discontinuous Galerkin (HDG) solver for the time-dependent compressible Euler and Navier-Stokes equations. In contrast to discontinuous Galerkin (DG) methods, the number of globally coupled degrees of freedom is usually tremendously smaller for HDG methods, as these methods can rely on hybridization. However, applying the method to a time-dependent problem amounts to solving a differential-algebraic nonlinear system of equations (DAE), rendering the problem extremely stiff. This implies that implicit time discretization should be used. Suited methods for the treatment of these DAE are, e.g., diagonally implicit Runge-Kutta (DIRK) methods, or the backward differentiation formulas (BDF). In order to solve a wide range of problems in an efficient manner, we employ adaptive time stepping using an embedded error estimator. Additionally, we investigate the use of artificial viscosity for shock capturing in this setting, and we propose a new strategy of coarsening the mesh using non-standard polygonal elements.