New optimized flux difference schemes for improving high-order weighted compact nonlinear scheme with applications

To improve the spectral characteristics of the high-order weighted compact nonlinear scheme (WCNS), optimized flux difference schemes are proposed. The disadvantages in previous optimization routines, i.e., reducing formal orders, or extending stencil widths, are avoided in the new optimized schemes by utilizing fluxes from both cell-edges and cell-nodes. Optimizations are implemented with Fourier analysis for linear schemes and the approximate dispersion relation (ADR) for nonlinear schemes. Classical difference schemes are restored near discontinuities to suppress numerical oscillations with use of a shock sensor based on smoothness indicators. The results of several benchmark numerical tests indicate that the new optimized difference schemes outperform the classical schemes, in terms of accuracy and resolution for smooth wave and vortex, especially for long-time simulations. Using optimized schemes increases the total CPU time by less than 4%.