Upwind Hybrid Spectral Difference Methods for Steady-State Navier–Stokes Equations
We propose an upwind hybrid spectral difference method for the steady-state Navier–Stokes equations. The (upwind) hybrid spectral difference method is based on a hybridization as follows: (1) an (upwind) spectral finite difference approximation of the Navier–Stokes equations within cells (the cell finite difference) and (2) an interface finite difference on edges of cells. The interface finite difference approximates continuity of normal stress on cell interfaces. The main advantages of this new approach are three folds: (1) they can be applied to non-uniform grids, retaining the order of convergence, (2) they are stable without using a staggered grid and (3) the schemes have an embedded static condensation property, hence, there is a big reduction in degrees of freedom in resulting discrete systems. The inf-sup condition is proved. Various numerical examples including the driven cavity problem with the Reynolds numbers, 5000–20,000, are presented.
Youngmok Jeon was supported by National Research Foundation of Korea (NRF-2015R1D1A1A09057935). Dongwoo Sheen was supported in part by National Research Foundation of Korea (NRF-2017R1A2B3012506 and NRF-2015M3C4A7065662).
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