Accuracy-preserving algorithm for difference schemes applied to non-uniform grids

Liu Jun; Jiang Jinyu

doi:10.7638/kqdlxxb-2025.0023

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Accuracy-preserving algorithm for difference schemes applied to non-uniform grids

更新时间：2026-07-11

- Accuracy-preserving algorithm for difference schemes applied to non-uniform grids
- Vol. 44, Issue 6, Pages: 25-33(2026)
- 作者机构：
  
  大连理工大学航空航天学院,大连,116024
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2025.0023
  CLC： V211.3
- Published：2026
- 稿件说明：
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刘君, 姜金裕. 差分格式应用于非均匀网格的精度保持算法[J]. 空气动力学学报, 2026, 44(6): 25−33.

Liu Jun, Jiang Jinyu. Accuracy-preserving algorithm for difference schemes applied to non-uniform grids[J]. 2026, 44(6): 25-33.
刘君, 姜金裕. 差分格式应用于非均匀网格的精度保持算法[J]. 空气动力学学报, 2026, 44(6): 25−33. DOI： 10.7638/kqdlxxb-2025.0023

Liu Jun, Jiang Jinyu. Accuracy-preserving algorithm for difference schemes applied to non-uniform grids[J]. 2026, 44(6): 25-33. DOI： 10.7638/kqdlxxb-2025.0023.

摘要

为解决有限差分格式在非均匀网格下精度下降的问题，本文通过研究一维非均匀网格中几何诱导误差的产生机制，构建仅具有一维非均匀性的网格模型，通过理论分析与数值实验揭示了传统格式精度损失的机理。理论研究发现：现有差分格式如一阶迎风、MUSCL、WENO等格式，在非均匀网格下难以达到理论设计精度，其根源在于坐标变换引入的高阶导数项对截断误差产生了不可控影响。基于有限差分法对导数逼近的本质目标，创新性提出精度保持算法。该方法通过加权融合不同方向差商项，修正传统格式在非均匀网格上的离散策略，以重构导数逼近模型来有效消除几何诱导误差。通过线性流与平方流为初场进行数值验证，结果表明：改进后的一阶格式在非均匀网格中可严格保持线性流场（误差量级为10

−17

），二阶格式可进一步保持平方流场（误差量级为10

−16

），本研究结果可为后续非均匀网格误差消减算法的完善提供新思路。

Abstract

To address the issue of accuracy degradation in finite difference schemes on non-uniform grids

this study investigated the mechanism behind geometry-induced errors in one-dimensional non-uniform grid configurations. A simplified grid model with only one-dimensional non-uniformity was constructed

and both theoretical analysis and numerical experiments were conducted to reveal the root causes of accuracy loss in traditional schemes. Theoretical findings indicated that existing difference schemes

such as first-order upwind

MUSCL and WENO schemes

failed to achieve their designed order of accuracy on non-uniform g

rids due to the uncontrolled influence of higher-order derivative terms introduced by coordinate transformations on the truncation error. Based on a re-examination of the fundamental goal of derivative approximation in finite difference methods

an innovative accuracy-preserving algorithm was proposed. This method reconstructed the derivative approximation model by combining directional difference quotients in a weighted manner

thereby correcting the discretization strategy of traditional schemes and effectively eliminating geometry-induced errors. Numerical validations using linear and quadratic flow fields as initial conditions demonstrated that the improved first-order scheme could strictly preserve linear flow fields (with error magnitude on the order of 10

−17

)

and the second-order scheme could further preserve quadratic flow fields (with error magnitude on the order of 10

−16

). The findings of this study offer new insights for the development of error-reduction algorithms on non-uniform grids.

关键词

Keywords

references

Chen Jianqiang, Ma Yankai, Min Yaobing, et al. Design and development of homogeneous hybrid solvers on National Numerical Windtunnel (NNW) PHengLEI[J]. Acta Aerodynamica Sinica, 2020, 38(6): 1103−1110,1102(in Chinese). doi: 10.7638/kqdlxxb-2020.0177

Chang S, Bai X, Zou D, et al. An adaptive discontinuity fitting technique on unstructured dynamic grids[J]. Shock Waves, 2019, 29(8): 1103−1115. doi: 10.1007/s00193-019-00913-3

Liu Jun, Chen Zedong, Chen Jie, et al. A surface extraction method of flow feature for shock-fitting scheme[J]. Acta Aerodynamica Sinica, 2019, 37(3): 373−381(in Chinese).

Samarskii A A, Radulescu V D. Theory of difference schemes[J]. Applied Mechanics Reviews, 2002, 55(2): B24−B25. doi: 10.1115/1.1451158

Trulio J, Trigger K R. Numerical solution of the one-dimensional Lagrangian hydrodynamic equations[R]. California, USA: Lawrence Radiation Lab. , 1961. doi: 10.2172/4782302

Thomas P, Lombard C. The geometric conservation law - A link between finite-difference and finite-volume methods of flow computation on moving grids[C]//Proc of the 11th Fluid and PlasmaDynamics Conference, Seattle, WA. Reston, Virginia: AIAA, 1978: 1208. doi: 10.2514/6.1978-1208

Thomas P D, Lombard C K. Geometric conservation law and its application to flow computations on moving grids[J]. AIAA Journal, 1979, 17(10): 1030−1037. doi: 10.2514/3.61273

Liu Jun, Han Fang. Body-fitted coordinate transformation for finite difference method[J]. Physics of Gases, 2018, 3(5): 18−29(in Chinese).

Liu Jun, Han Fang, Xia Bing. Mechanism and algorithm for geometric conservation law in finite difference method[J]. Acta Aerodynamica Sinica, 2018, 36(6): 917−926(in Chinese). doi: 10.7638/kqdlxxb-2018.0034

Han F, Xu C G, Liu J. Improvement to a general methodology for free-stream preservation on curvilinear grids[J]. Physics of Fluids, 2022, 34(11): 116111. doi: 10.1063/5.0120313

Astoul T , Wissocq G , Boussuge J F , et al. Lattice Boltzmann method for computational aeroacoustics on non-uniform meshes: A direct grid coupling approach[J]. Journal of Computational Physics, 2021, 447. 110667. doi: 10.1016/j.jcp.2021.110667

Li Xinliang. Direct numerical simulation of channel turbulence [D]. Beijing: Institute of Mechanics, Chinese Academy of Sciences, 2001(in Chinese).

Liu Jun, Wei Yanxin, Han Fang. Coordinate transformation induced errors of finite difference method[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(6): 124397(in Chinese). doi: 10.7527/S1000-6893.2020.24397

Wang R, Feng H, Spiteri R J. Observations on the fifth-order WENO method with non-uniform meshes[J]. Applied Mathematics and Computation, 2008, 196(1): 433−447. doi: 10.1016/j.amc.2007.06.024

Huang W F, Ren Y X, Jiang X. A simple algorithm to improve the performance of the WENO scheme on non-uniform grids[J]. Acta Mechanica Sinica, 2018, 34(1): 37−47. doi: 10.1007/s10409-017-0715-2

Hairer E, Wanner G. Solving ordinary differential equations II: stiff and differential-algebraic problems[M]. Berlin: Springer Basel, 1996. doi: 10.1007/978-3-642-05221-7

Shu C W. High order weighted essentially nonoscillatory schemes for convection dominated problems[J]. SIAM Review, 2009, 51(1): 82−126. doi: 10.1137/070679065

Liu Jun, Han Fang, Wei Yanxin. MUSCL and WENO schemes problems generated by dimension splitting approach[J]. Acta Aeronautica et Astronautica Sinica, 2022, 43(3): 125009(in Chinese). doi: 10.7527/S1000-6893.2021.25009

Persson P O, Peraire J. Curved mesh generation and mesh refinement using Lagrangian solid mechanics[C]//Proc of the 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition, Orlando, Florida. Reston, Virginia: AIAA, 2009: 949. doi: 10.2514/6.2009-949

Zhao Wei, Chen Jiangtao, Xiao Wei, et al. Advances in the key technologies of verification and validation system of National Numerical Windtunnel project[J]. Acta Aerodynamica Sinica, 2020, 38(6): 1165−1172(in Chinese). doi: 10.7638/kqdlxxb-2020.0138

Jiang G S, Shu C W. Efficient implementation of Weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126(1): 202−228. doi: 10.1006/jcph.1996.0130

Van Leer B. Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method[J]. Journal of Computational Physics, 1979, 32(1): 101−136. doi: 10.1016/0021-9991(79)90145-1

Zhang Hanxin. Non-oscillatory and non-free-parameter dissipation difference scheme[J]. Acta Aerodynamica Sinica, 1988, 6(2): 143−165(in Chinese).

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