52nd AIAA Aerospace Sciences Meeting, 13-17 January 2014, National Harbor, Maryland A Hybridized Discontinuous Galerkin Method for Three-Dimensional Compressible Flow Problems Michael Woopen ⇤ , Aravind Balan ⇤ , and Georg May † Graduate School AICES, RWTH Aachen University, 52062 Aachen, Germany We present a hybridized discontinuous Galerkin method for three-dimensional flow prob- lems. As an implementation technique hybridization is a classic paradigm for dual-mixed finite element discretizations. Hybridization of finite element discretizations has the main advantage, that the resulting set of algebraic equations has globally coupled degrees of freedom only on the skeleton of the numerical mesh. Solving for these thus involves the solution of a potentially much smaller system. This not only reduces storage requirements, but also allows for a faster solution with iterative solvers. The accuracy of the method has been validated with a scalar convection-diffusion test case. Results are shown for external, compressible flow. I. Introduction Within the last decades, several discretization techniques for partial differential equations have evolved. Two popular families, finite element and finite volume methods, are constantly utilized in industry. However, the requirements on these techniques are continuously increasing, necessitating further work on more accurate and efficient methodologies. In view of the advantages and disadvantages of these two methods, the so-called discontinuous Galerkin methods have attracted interest. Despite their popular advantages — high-order accuracy on unstructured meshes, a variational setting, and local conservation, just to name a few — they introduce a large number of degrees of freedom and they lead to large stencils due to an increased coupling across element interfaces (we refer to Cockburn [4] for an introduction into DG methods for various kinds of applications). In order to avoid these disadvantages, a technique called hybridization may be utilized. Here, the globally coupled unknowns have support on the mesh skeleton, i.e. the element interfaces, only. As a result, the global system is decreased in size and improved in terms of sparsity at the same time. Nguyen et al. devised a hybridizable discontinuous Galerkin method for linear convection-diffusion equa- tions in [7] and extended it to nonlinear convective fluxes in [8]. They observed that the approximations for the scalar variable and the flux converge with the optimal order in the L 2 -norm. Peraire et al. then applied this method to the compressible Euler and Navier-Stokes in [9]. They showed that their method produces optimal convergence rates for both the conserved quantities as well as the viscous stresses and the heat fluxes. In [12], Sch¨ utz and May presented a discretization method for nonlinear convection-diffusion equations including the compressible Navier-Stokes equations. The method is based on a discontinuous Galerkin discretization for convection terms, and a mixed method using H(div) spaces for the diffusive terms. Fur- thermore, hybridization is used to reduce the number of globally coupled degrees of freedom. A comparison of hybridized and standard DG methods is presented by Woopen et al. [13]. HDG is reported to be su- perior in terms of the efficiency, for the different test cases presented. In the present work, we extend our two-dimensional HDG scheme to three dimensions. This paper is structured as follows. We will briefly cover the governing equations, namely the compressible Euler equations in Sec. II. After that we introduce our discretization and describe the concept of hybridization in Sec. III. Next, we validate our method with a scalar test case of which the analytical solution is known a * Graduate Student † Junior Professor Copyright c  2014 by Michael Woopen. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 1 of 15 American Institute of Aeronautics and Astronautics Paper 2014-0938