High-Order Discontinuous Galerkin Methods for CFD

In recent years it has become clear that the current computational methods for scientific and engineering phenomena are inadequate for many challenging problems. Examples of these problems are wave propagation, turbulent fluid flow, as well as problems involving nonlinear interactions and multiple scales. This has resulted in a significant interest in so-called high-order accurate methods, which have the potential to produce more accurate and reliable solutions. A number of high-order numerical methods appropriate for flow simulation have been proposed, including finite difference methods, high-order finite volume methods, stabilized finite element methods, Discontinuous Galerkin (DG) methods, hybridized DG methods, and spectral element/difference methods. All of these methods have advantages in particular situations, but for various reasons most general purpose commercial-grade simulation tools still use traditional low-order methods. Much of the current research is devoted to the discontinuous Galerkin method. This is partly because of its many attractive properties, such as a rigorous mathematical foundation, the ability to use arbitrary orders of discretization on general unstructured simplex meshes, and the natural stability properties for convective-diffusive operators. In this chapter, we describe our work on efficient DG methods for unsteady compressible flow applications, including deformable domains and turbulent flows.