American Institute of Aeronautics and Astronautics 1 An Implicit Discontinuous Galerkin Method for the Unsteady Compressible Navier-Stokes Equations Hong Luo 1 and Hidehiro Segawa 2 North Carolina State University, Raleigh, NC 27695, USA Miguel R. Visbal 3 Air Force Research Laboratory, Wright-Patterson AFB, OH 45433, USA A high-order implicit discontinuous Galerkin method is developed for the time-accurate solutions to the compressible Navier-Stokes equations. The spatial discretization is carried out using a high order discontinuous Galerkin method, where polynomial solutions are represented using a Taylor basis. A second order implicit method is applied for temporal discretization to the resulting ordinary differential equations. The resulting nonlinear system of equations is solved at each time step using a pseudo-time marching approach. A newly developed fast, p-multigrid is then used to obtain the steady state solution to the pseudo-time system. The developed method is applied to compute a variety of unsteady viscous flow problems. The numerical results obtained indicate that the use of this implicit method leads to orders of improvements in performance over its explicit counterpart, while without significant increase in memory requirements. I. Introduction hile DG was originally introduced by Reed and Hill 1 for solving the neutron transport equation back in 1973, major interest did not focus on it until the nineties 2-5 . Nowadays, it is widely used in the computational fluid dynamics, computational aeroacoustics, and computational electromagnetics, to name just a few 6-21 . What is known so far about this method offers a tantalizing glimpse of its full potential. Indeed, what sets this method apart from the crowd is many attractive features it possesses: 1) It has several useful mathematical properties with respect to conservation, stability, and convergence; 2) The method can be easily extended to higher-order (>2nd) approximation; 3) The method is well suited for complex geometries since it can be applied on unstructured grids. In addition, the method can also handle non-conforming elements, where the grids are allowed to have hanging nodes; 4) The method is highly parallelizable, as it is compact and each element is independent. Since the elements are discontinuous, and the inter-element communications are minimal, domain decomposition can be efficiently employed. The compactness also allows for structured and simplified coding for the method; 5) It can easily handle adaptive strategies, since refining or coarsening a grid can be achieved without considering the continuity restriction commonly associated with the conforming elements. The method allows easy implementation of hp-refinement, for example, the order of accuracy, or shape, can vary from element to element. 6) It has the ability to compute low Mach number flow problems without any special treatment. In recent years, significant progress has been made in developing numerical algorithms for solving the compressible Euler and Navier-Stokes equations using discontinuous Galerkin methods. Most of these numerical methods are based on the semi-discrete approach: discontinuous Galerkin finite element methods are used for the spatial discretization, rendering the original partial differential equations (PDE) into a system of ordinary differential equations (ODE) in time. After constructing the ODE system, a time-stepping strategy is used to advance the solution in time. Usually, explicit temporal discretizations such as multi-stage Runge-Kutta schemes are used to integrate the semi-discrete system in time. In general, explicit schemes and their boundary conditions are easy to implement, vectorize and parallelize, and require only limited memory storage. However, for large-scale problems and especially for the higher-order DG solutions, the rate of convergence slows down dramatically, resulting in 1 Associate Professor, Department of Mechanical and Aerospace Engineering, Senior Member AIAA. 2 Ph.D. student, Department of Mechanical and Aerospace Engineering, Student Member AIAA. 3 Team Leader, Computational Sciences Branch, Associate Fellow AIAA. W 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5 - 8 January 2009, Orlando, Florida AIAA 2009-951 Copyright © 2009 by authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.