Spectral/hp methods for viscous compressible flows on unstructured 2D meshes

In this paper we describe the foundation of a spectral/hp method suitable for simulating viscous compressible flows with shocks on standard unstructured meshes. It is based on a discontinuous Galerkin formulation for the hyperbolic contributions combined with a mixed Galerkin formulation fcr 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 coordinate system that retains a tensor product property and accurate numerical quadrature. The formulation is conservative, and monotonicity is enforced by high-order limiters and by appropriately lowering the basis order around discontinuities. Convergence results are shown for benchmark solutions of the advection, Euler, and Navier-Stokes equations that demonstrate exponential convergence of the new method. Flow simulations for subsonic and supersonic flows are also presented that demonstrate discretization flexibility using h - p type refinement. Unlike other high-order methods the new method uses standard finite volume meshes consisting of arbitrary triangulizations. (C) 1998 Academic Press.