高阶精度有限差分格式的全局守恒特性初探

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

  • 摘要: 高阶精度有限差分格式的守恒特性一直都饱受质疑,这限制了其在复杂工程问题中的应用。基于对称守恒网格导数方法(symmetric conservation metric method, SCMM)的高阶精度WCNS(weighted compact nonlinear schemes)格式,具有与有限体积方法相媲美的几何守恒特性,但其全局守恒特性却未能彻底解决。本文基于等距网格上的全场一致高阶精度WCNS格式,将整个计算区域内所有求解点上的积分权值作为未知量,以满足全局守恒为条件,直接数值解出所有求解点上的积分权值,并结合理论分析给出积分权值应该满足的4个约束条件,然后通过针对性的数值实验分别验证了4个约束条件的合理性,并进一步研究了全局守恒特性对高阶精度差分格式模拟结果的影响。研究结果可用于指导适用于工程实际问题模拟高阶精度边界差分格式和对应的数值积分方法的设计。

     

    Abstract: The conservation properties of high-order finite difference schemes have consistently been questioned, limiting their application in complex engineering problems. Based on the Symmetrical 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 computational 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.

     

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