Optimized high-order schemes with damping terms for discretization of viscous fluxes in the framework of weighted compact non-linear schemes

Compared with the inviscid flux discretization, the viscous flux discretization has attracted less attention for the high-order finite difference schemes. In general, the central scheme is employed for the discretization of viscous flux. In this paper, we optimize the central schemes by adding up a damping term with free parameters in the framework of weighted compact non-linear schemes. The optimization is for both the straight and mixed second derivative terms present in the viscous flux of Navier–Stokes equations. By using multi-dimensional Taylor series expansion and Fourier spectral analysis, free parameters are determined. Compared with the original central schemes, optimized schemes demonstrate superior spectral properties, especially in the range of high wave number. Numerical tests are conducted to verify the effectiveness of optimized schemes. Results show that the designed sixth-order of accuracy is achieved, and the optimized schemes with damping terms are capable of suppressing non-physical small-scale structures and oscillations, for both smooth and non-smooth flowfields.