The conservation properties of high-order finite difference schemes have consistently been questioned, limiting their application in complex engineering problems. Based on the symmetric conservative metric method (SCMM), high-order weighted compact nonlinear schemes (WCNS) exhibit geometric conservation properties comparable to those of the finite volume method. However, their global conservation properties have not been fully addressed. This paper introduces a uniformly high-order WCNS scheme on equidistant grids, where the integral weights at all solution points are treated as unknowns and directly solved numerically to ensure exact satisfaction of global conservation. Theoretical analysis provides four constraint conditions that the integral weights should satisfy, which are then validated through targeted numerical experiments. Additionally, we examine the impact of the global conservation property on the simulation results of high-order finite difference schemes. The findings of this study can inform the design of high-order boundary difference schemes and numerical integration methods for engineering applications.

After a brief introduction to the historical development of nonlinear schemes for capturing discontinuities, this paper delves into the research on selecting candidate stencils, calculating smoothness indicators and their relationships with nonlinear weighting functions. This review is centered on the fifth order WENO scheme, serving as a representative example of nonlinear weighted schemes. It highlights the crucial requirements for the leading term order of nonlinear weights to preserve the accuracy of these schemes, especially when dealing with candidate stencils of varying widths. This paper underscores the significance of smoothness indicator calculations in ensuring both the accuracy and efficiency of the scheme. Finally, it offers insights and suggestions for future research aimed at further enhancing nonlinear weighted schemes that utilize candidate stencils with non-uniform widths.

Compressible turbulent flow widely exists in the aerospace field, and the coexistence of shock discontinuity and multi-scale turbulence in the flow field poses a challenge for high-precision numerical simulation. To improve the accuracy of complex turbulence simulation, in the framework of the Weighted Compact Nonlinear Scheme (WCNS), a WCNS-T scheme based on the Targeted Essentially Non-Oscillatory (TENO) weighting strategy is developed, and a smooth S-TENO nonlinear weighting method and its WCNS-ST scheme are further proposed from the perspectives of both numerical convergence and accuracy. The developed WCNS-ST scheme is tested for its ability to capture discontinuous and high-frequency waves through spectral analysis, one-dimensional Lax and Osher-Shu examples, and two-dimensional dual Mach reflection examples. Based on further numerical tests of steady laminar flow past a circular cylinder and unsteady self-adaptive turbulence eddy simulation of a 30P30N multi-element airfoil, the convergence characteristics of the original TENO weighting and the current S-TENO weighting in viscous complex flows, as well as their impact on numerical calculation results, are compared. The results show that the proposed S-TENO nonlinear weighting maintains the dispersion and dissipation characteristics of the original TENO nonlinear weighting and its ability to capture shock waves, significantly improving the problem of excessive nonlinear dissipation in WCNS-JS and WCNS-Z schemes; meanwhile, S-TENO nonlinear weighting significantly eases the convergence difficulties caused by weight discontinuity in the original TENO nonlinear weighting. The above results indicate that the proposed smooth S-TENO nonlinear weighting and its WCNS-ST scheme take into account both accuracy and convergence in numerical simulations of complex flow configurations, making them more suitable for complex engineering applications.