The optimization of the compact upwind scheme for incompressible flow

Different compact upwind schemes have been developed and used to numerically approximate a convection term in the Navier–Stokes equation. With different point stencils, the compact upwind schemes are mainly classified as the central, the function-biased, and the derivative-biased compact upwind schemes. They have different numerical characteristics. In this paper, by using Fourier analysis and numerical test, it is found that the function-biased compact upwind schemes have better resolution properties than the derivative-biased compact upwind schemes. Furthermore, an optimization method named dispersion-dissipation-balancing (DDB) optimization is proposed, by which better spectral resolution of these schemes is obtained by optimizing coefficients of these schemes based on the balance between a dispersion error and a dissipation error. Compared with the popular dispersion-relation-preserving (DRP) optimization method, the schemes optimized by the DDB method have proper dispersion and dissipation errors. They eliminate both the nonphysical oscillations and spurious vortices in the numerical case of the double shear layers flow. In addition, the central compact upwind scheme optimized by the DDB method (OCCUS_DDB) has the best performance among the schemes studied in this paper.