高阶精度有限差分格式的全局守恒特性初探

闵耀兵; 谢宁; 马燕凯; 燕振国; 邓小刚

doi:10.7638/kqdlxxb-2024.0111

欢迎来到能源电力期刊网！

English Version 期刊登录

您当前的位置：

首页 >

文章列表页 >

高阶精度有限差分格式的全局守恒特性初探

研究论文 | 更新时间：2026-05-28

- 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
- 空气动力学学报 2026年44卷第1期页码：19-29
- 作者机构：
  
  1. 中国空气动力研究与发展中心计算空气动力研究所,绵阳,621000
  2. 军事科学院,北京,100071
- 作者简介：
  
  闵耀兵（1982—），男，湖北黄冈人，博士，高级工程师，研究方向：计算流体力学. E-mail：minyb@126.com
  马燕凯*，助理研究员，研究方向：高阶精度有限差分方法及其应用. E-mail：mayankai_cardc@163.com
- 基金信息：
- DOI：10.7638/kqdlxxb-2024.0111
  中图分类号： V211.3;O13
- 纸质出版：2026
- 稿件说明：
移动端阅览
闵耀兵, 谢宁, 马燕凯, 等. 高阶精度有限差分格式的全局守恒特性初探[J]. 空气动力学学报, 2026, 44(1): 19−29.

Min Yaobing, Xie Ning, Ma Yankai, et al. Exploration of global conservation properties in high-order finite difference schemes[J]. 2026, 44(1): 19-29.
闵耀兵, 谢宁, 马燕凯, 等. 高阶精度有限差分格式的全局守恒特性初探[J]. 空气动力学学报, 2026, 44(1): 19−29. DOI： 10.7638/kqdlxxb-2024.0111

Min Yaobing, Xie Ning, Ma Yankai, et al. Exploration of global conservation properties in high-order finite difference schemes[J]. 2026, 44(1): 19-29. DOI： 10.7638/kqdlxxb-2024.0111.

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，但其全局守恒特性问题却未能彻底解决。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 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.

关键词

Keywords

references

Wang Z J. High-order methods for the Euler and Navier–Stokes equations on unstructured grids[J]. Progress in Aerospace Sciences, 2007, 43(1-3): 1−41. doi: 10.1016/j.paerosci.2007.05.001

Wang Z J, Fidkowski K, Abgrall R, et al. High-order CFD methods: current status and perspective[J]. International Journal for Numerical Methods in Fluids, 2013, 72(8): 811−845. doi: 10.1002/fld.3767

Xu Dan. Research on high-precision finite difference method satisfying geometric conservation law and its application in large eddy simulation[D]. Changsha: National University of Defense Technology, 2017(in Chinese).

Deng Xiaogang, Min Yaobing, Mao Meiliang, et al. Further studies on Geometric Conservation Law and applications to high-order finite difference schemes with stationary grids[J]. Journal of Computational Physics, 2013, 239: 90−111. doi: 10.1016/j.jcp.2012.12.002

Deng Xiaogang, Zhu Huajun, Min Yaobing, et al. High-order finite difference schemes based on symmetric conservative metric method: decomposition, geometric meaning and connection with finite volume schemes[J]. Advances in Applied Mathematics and Mechanics, 2020, 12(2): 436−479. doi: 10.4208/aamm.oa-2017-0243

Liao F, He G W. High-order adapter schemes for cell-centered finite difference method[J]. Journal of Computational Physics, 2020, 403: 109090. doi: 10.1016/j.jcp.2019.109090

Perot J B. Discrete conservation properties of unstructured mesh schemes[J]. Annual Review of Fluid Mechanics, 2011, 43: 299−318. doi: 10.1146/annurev-fluid-122109-160645

Deng Xiaogang, Zhang Hanxin. Developing high-order weighted compact nonlinear schemes[J]. Journal of Computational Physics, 2000, 165(1): 22−44. doi: 10.1006/jcph.2000.6594

Svärd M, Nordström J. Review of summation-by-parts schemes for initial–boundary-value problems[J]. Journal of Computational Physics, 2014, 268: 17−38. doi: 10.1016/j.jcp.2014.02.031

Del Rey Fernández D C, Hicken J E, Zingg D W. Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations[J]. Computers Fluids, 2014, 95: 171−196. doi: 10.1016/j.compfluid.2014.02.016

Morinishi Y, Lund T S, Vasilyev O V, et al. Fully conservative higher order finite difference schemes for incompressible flow[J]. Journal of Computational Physics, 1998, 143(1): 90−124. doi: 10.1006/jcph.1998.5962

Morinishi Y. Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows[J]. Journal of Computational Physics, 2010, 229(2): 276−300. doi: 10.1016/j.jcp.2009.09.021

Pirozzoli S. Generalized conservative approximations of split convective derivative operators[J]. Journal of Computational Physics, 2010, 229(19): 7180−7190. doi: 10.1016/j.jcp.2010.06.006

Van’T Hof B, Veldman A E P. Mass, momentum and energy conserving (MaMEC) discretizations on general grids for the compressible Euler and shallow water equations[J]. Journal of Computational Physics, 2012, 231(14): 4723−4744. doi: 10.1016/j.jcp.2012.03.005

Coppola G, Capuano F, De Luca L. Discrete energy-conservation properties in the numerical simulation of the navier–stokes equations[J]. Applied Mechanics Reviews, 2019, 71(1): 010803. doi: 10.1115/1.4042820

Coppola G, Capuano F, Pirozzoli S, et al. Numerically stable formulations of convective terms for turbulent compressible flows[J]. Journal of Computational Physics, 2019, 382: 86−104. doi: 10.1016/j.jcp.2019.01.007

Turner J M, Haeri S, Kim J W. Improving the boundary efficiency of a compact finite difference scheme through optimising its composite template[J]. Computers Fluids, 2016, 138: 9−25. doi: 10.1016/j.compfluid.2016.08.007

Brady P T, Livescu D. High-order, stable, and conservative boundary schemes for central and compact finite differences[J]. Computers Fluids, 2019, 183: 84−101. doi: 10.1016/j.compfluid.2018.12.010

Deng Xiaogang, Chen Yamin. A novel strategy for deriving high-order stable boundary closures based on global conservation, I: Basic formulas[J]. Journal of Computational Physics, 2018, 372: 80−106. doi: 10.1016/j.jcp.2018.06.012

Qin Jiaxian, Chen Yaming, Deng Xiaogang. On the role of global conservation property for finite difference schemes[J]. Journal of Computational Physics, 2021, 440: 110437. doi: 10.1016/j.jcp.2021.110437

Zhong Xiaolin, Tatineni M. High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition[J]. Journal of Computational Physics, 2003, 190(2): 419−458. doi: 10.1016/s0021-9991(03)00282-1

Shukla R K, Zhong Xiaolin. Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation[J]. Journal of Computational Physics, 2005, 204(2): 404−429. doi: 10.1016/j.jcp.2004.10.014

Hermanns M, Hernández J A. Stable high-order finite-difference methods based on non-uniform grid point distributions[J]. International Journal for Numerical Methods in Fluids, 2008, 56(3): 233−255. doi: 10.1002/fld.1510

Gustafsson B. The convergence rate for difference approximations to mixed initial boundary value problems[J]. Mathematics of Computation, 1975, 29(130): 396−406. doi: 10.1090/s0025-5718-1975-0386296-7

Gustafsson B. The convergence rate for difference approximations to general mixed initial-boundary value problems[J]. SIAM Journal on Numerical Analysis, 1981, 18(2): 179−190. doi: 10.1137/0718014

Min Yaobing, Ma Yankai, Li Song. Accuracy analysis of numerical error with statistical forms in CFD[J]. Acta Aeronauticaet Astronautica Sinica, 2020, 41(4): 123554(in Chinese). doi: 10.7527/S1000-6893.2019.23554

Abe Y, Nonomura T, Iizuka N, et al. Geometric interpretations and spatial symmetry property of metrics in the conservative form for high-order finite-difference schemes on moving and deforming grids[J]. Journal of Computational Physics, 2014, 260: 163−203. doi: 10.1016/j.jcp.2013.12.019

浏览量

下载量

CSCD

文章被引用时，请邮件提醒。

提交

工具集

关联资源

风扇叶片颗粒侵蚀特性的多因素分析与模拟

个体间距对多扑翼气动噪声的影响特性

热防护材料动态烧蚀形貌结构光视觉测量技术

交错型CPR方法的保正实现策略及其对比研究

离心通风机降噪研究进展