有限容积法离散方程截断误差本质及其来源  

Essence and composition of truncation errors of discrete equations based on the finite volume method

作　　者：宇波[1] 焦开拓 陈宇杰 韩东旭 李敬法 孙东亮[1] Bo Yu;Kaituo Jiao;Yujie Chen;Dongxu Han;Jingfa Li;Dongliang Sun(School of Mechanical Engineering,Beijing Institute of Petrochemical Technology,Beijing 102617,China;State Key Laboratory of Multiphase Flow in Power Engineering,Xi’an Jiaotong University,Xi’an 710049,China)

机构地区：[1]北京石油化工学院机械工程学院,北京102617 [2]西安交通大学动力工程多相流国家重点实验室,西安710049

出　　处：《科学通报》2023年第10期1266-1280,共15页Chinese Science Bulletin

基　　金：国家自然科学基金(51904031)资助。

摘　　要：本文从有限容积法的原理出发,指出有限容积法离散方程截断误差的本质为离散表达式与守恒型积分方程的近似程度.基于该思想,推导了有限容积法离散方程截断误差表达式,该表达式与有限差分法截差表达式有显著的不同;并指出有限容积法中涉及的截断误差主要包含3个层次,分别为由界面上值的近似、界面通量的近似和离散方程的近似带来的截断误差.有限容积法界面通量截断误差包含法向和切向两部分,有限差分法不存在界面未知值近似和切向截断误差,因而即使对于常物性二维/三维问题,两者截差表达式也不相同.与有限差分法相比,有限容积法误差来源较多,单纯提高待求变量的离散精度并不能有效地提高离散方程的整体精度.此外,通过理论推导和数值试验证明了对于规则区域正交非均分网格,计算区域采用内节点法离散的计算精度总体上优于外节点法.Based on the principle of the finite volume method,the essence of truncation errors of discrete equations is revealed that represents the approximation degree to the conservative integral equation instead of the differential equation at the node or the average value over the control volume.This idea differs from that the truncation error expresses the approximation degree between the discrete expression and the differential equation in the finite difference method.The truncation error expression of discrete equations obtained by the finite volume method is derived,explaining that the truncation errors mainly involve three levels.The first level comes from the variables at the cell face,including the target variable and its first-order derivative,velocity,diffusion coefficient,etc.The second level originates from the flux at the cell face,consisting of the normal and tangential truncation errors:The former is caused by the approximations of the target variable and its first derivative,mass flow,and diffusion coefficient at the cell interface;the latter is induced by taking the value at one point as the average value of the cell face.The third level derives from discrete equations,and truncation errors include the convection and diffusion fluxes errors at the cell face and source term errors of the control volume.It can be found that there is an essential difference between the finite volume method and the finite difference method in truncation errors since there are no approximations of unknown parameters and tangential truncation errors at the cell face for the latter.Moreover,the truncation errors of the finite difference method are derived based on the grid node.Therefore,even for ideal twodimensional or three-dimensional problems with constant physical properties,the truncation error expressions of these two methods are different.Because there are many sources of errors for the finite volume method,the overall accuracy of discrete equations cannot be improved by only raising the discrete accuracy of the target

关 键 词：有限容积法 有限差分法 离散方程 截断误差 

分 类 号：O241.8[理学—计算数学]