A discontinuous Galerkin method for the Navier–Stokes equations

✍ Scribed by Igor Lomtev; George Em Karniadakis

• Publisher: John Wiley and Sons
• Year: 1999
• Tongue: English
• Weight: 526 KB
• Volume: 29
• Category: Article
• ISSN: 0271-2091
• DOI: 10.1002/(sici)1097-0363(19990315)29:5<587::aid-fld805>3.0.co;2-k

✦ Synopsis

The foundations of a new discontinuous Galerkin method for simulating compressible viscous flows with shocks on standard unstructured grids are presented in this paper. The new method is based on a discontinuous Galerkin formulation both for the advective and the diffusive contributions. High-order accuracy is achieved by using a recently developed hierarchical spectral basis. This basis is formed by combining Jacobi polynomials of high-order weights written in a new co-ordinate system. It retains a tensor-product property, and provides accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by appropriately lowering the basis order 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 subsonic, transonic and supersonic flows are also presented that demonstrate discretization flexibility using hp-type refinement. Unlike other high-order methods, the new method uses standard finite volume grids consisting of arbitrary triangulizations and tetrahedrizations.