A discontinuous Galerkin method for the Navier-Stokes equations

The foundations of a new discontinuous Galerkin method for simulating compr essible viscous flows with shocks on standard unstructured grids are presen ted in this paper. The new method is based on a discontinuous Galerkin form ulation both for the advective and the diffusive contributions. High-order accuracy is achieved by using a recently developed hierarchical spectral ba sis. This basis is formed by combining Jacobi polynomials of high-order wei ghts written in a new co-ordinate system. It retains a tenser-product prope rty, and provides accurate numerical quadrature. The formulation is conserv ative, and monotonicity is enforced by appropriately lowering the basis ord er and performing h-refinement around discontinuities. Convergence results are shown for analytical two- and three-dimensional solutions of diffusion and Navier-Stokes equations that demonstrate exponential convergence of the new method, even for highly distorted elements. Flow simulations for subso nic, transonic and supersonic flows are also presented that demonstrate dis cretization flexibility using hp-type refinement. Unlike other high-order m ethods, the new method uses standard finite volume grids consisting of arbi trary triangulizations and tetrahedrizations. Copyright (C) 1999 John Wiley & Sons, Ltd.