Exploration of global conservation properties in high-order finite difference schemes

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.