A Class of Third-Order Compact Upwind Schemes for Compressible Flow with Shocks and Vortices

Abstract

We present a class of third-order upwind compact schemes and develop an implementation strategy for un-steady compressible flow applications with strong shocks and vortices. Two adjustable parameters are left for the user to tune the scheme for specific applications. A third-order upwind scheme reported by Tolstykh and termed CUD-3 (High accuracy non-central compact difference schemes for fluid dynamic applications, World Scientific) turns out to be a particular case of the presented class of schemes. The aforementioned adjustable parameters allow us to derive schemes that perform better than the CUD-3. A large number of compressible solvers in use today for applications involving sharp gradients in the flow field employs flux limiters. We pose our compact scheme as a cell-face variable interpolation scheme so that available limiters may be applied during construction of the cell-face fluxes. Solvers based on popular schemes such as the AUSM class of schemes or the CUSP will require minimal modification to incorporate the proposed flux reconstruction method using compact upwinding. A number of test cases in one dimensions are then solved to show the usefulness of the proposed scheme and its implementation strategy.