PhysicsUpwind Compact and Explicit High - Order Finite Di erenceSchemes for Direct Numerical Simulation of High - Speed

Direct numerical simulation of transitional and turbulent hypersonic boundary layers using the Navier-Stokes equations requires high-order accurate numerical methods to resolve a wide range of time and length scales. Compact or explicit nite di erence methods used for such simulation have mainly been central di erence schemes containing only phase errors without any numerical dissipation. Central di erence schemes of fourth or higher orders, however, are often unstable when they are coupled with high-order one-sided nite-di erence boundary schemes. In addition, extra ltering procedures, which are equivalent to adding numerical dissipation, are often required for central schemes to control the aliasing errors and to stabilize the computations. This paper presents and analyzes a family of upwind compact and explicit nite di erence schemes of third, fth, and seventh orders and their stable boundary schemes. In each of the new upwind schemes, central grid stencil is used with built-in implicit numerical dissipation controlled by a free damping parameter. Fourier analysis is used to analyze the dissipation and phase errors of the schemes. The dissipation errors are designed to be smaller or comparable to the phase errors for well resolved low wavenumber modes and to damp out unresolved higher wavenumber modes. The eigenvalue spectrums of spatial approximations with boundary conditions are computed to test the asymptotic stability of the schemes. It is found that the new high-order upwind schemes help to stabilize the overall schemes when they are coupled with high-order boundary closures. The accuracy and stability of the new high-order upwind schemes are con rmed by numerical experiments. 3