Development of high-order weighted compact schemes with various difference methods

This paper is concerned with the development of high-order weighted compact schemes based on linear central compact schemes (Liu et al., 2013). For the purpose of capturing strong discontinuities, the high-order upwind weighted nonlinear interpolations are combined with the central compact schemes and cell-centered compact schemes to achieve two different weighted compact schemes, called the CCSSR-W and CCCSSR-W schemes respectively. The CCSSR-W schemes are developed up to ninth order, and four different nonlinear weights of CCSSR-W schemes are investigated in detail. As an improvement of the CCSSR-HW scheme (Liu et al., 2015), the hybrid weighted nonlinear interpolation is substituted into the cell-centered compact scheme to obtain the CCCSSR-HW scheme. For all the proposed schemes, the flux vector splitting method is carried out by coupling with characteristic projections on the conservative fluxes. Through the systematic accuracy tests, spectral analysis and numerical experiments, the results demonstrate that the higher order CCSSR-W scheme is more accurate and effective than the lower order ones. The CCCSSR-W scheme has better accuracy and resolution than the original WENO scheme. The CCCSSR-HW scheme has higher resolution and lower dissipation than the CCSSR-HW scheme, and even better than the higher order CCSSR-W schemes.