A Space-Time Discontinuous Galerkin Method for Navier-Stokes with Recovery.

This thesis tells the story of two new members of a new generation of discontinuous Galerkin (DG) methods. Although DG, as a spatial discretization, has exhibited unparalleled success in handling advection- dominated problems on unstructured grids, DG's development in diffusion and time-marching schemes has lagged far behind in comparison. The first part of this thesis contributes to a newly introduced diffusion scheme, and the second part to an arbitrarily high-order explicit time-marching scheme. Interface-centered recovery-based DG (RDG) was first introduced by Van Leer and Nomura (2005) as a new DG discretization for diffusion. The recovery concept is attractive because it is easily understood and can be extended beyond diffusion. The idea is to recover a smooth solution across a cell interface that shares all moments with the discontinuous solutions in the two cells adjacent to that interface. RDG proves to be a scheme of fine pedigree: its order of accuracy, 3p + 1 for odd p (p is the order of the solution polynomial), and 3p + 2 for even p, is unmatched on a Cartesian grid by any existing method for diffusion. For unstructured grids, the order of accuracy is reduced to the standard p+1, but RDG still allows of much larger time steps due to smaller eigenvalues. The extension to nonlinear problems with cross-derivative terms and two-dimensional structured and unstructured grids is described. Numerical tests for a range of diffusion equations, including Navier-Stokes, conrm the high-order accuracy of RDG. Van Leer (1977) coupled spatial and temporal operators to produce an exact shift in the solution for advection at a Courant-Friedrich-Lewy (CFL) number of unity. Huynh (2005) successfully implemented this concept in the "moment scheme" (p = 1) for the Euler equations. We extend this method to arbitrarily high order, and incorporate RDG for handling diffusion. The method is currently called Hancock-Huynh discon- tinuous Galerkin (HH-DG). We have tested it on Euler problems and linear advection-diffusion problems.Ph.D.Aerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86313/1/khlo_1.pd