Geometrically-Induced Errors in the Finite Difference Method for Solving Heat Conduction Equations

LIU Jun; LIU Guangying; XU Chunguang

doi:10.21656/1000-0887.460061

Volume 47 Issue 2

Feb. 2026

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LIU Jun, LIU Guangying, XU Chunguang. Geometrically-Induced Errors in the Finite Difference Method for Solving Heat Conduction Equations[J]. Applied Mathematics and Mechanics, 2026, 47(2): 219-229. doi: 10.21656/1000-0887.460061

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Geometrically-Induced Errors in the Finite Difference Method for Solving Heat Conduction Equations

doi: 10.21656/1000-0887.460061

1.
Faculty of Mechanical Engineering and Mechanics, Ningbo University, Ningbo, Zhejiang 315211, P.R.China
2.
School of Aeronautics and Astronautics, Sun Yat-sen University, Guangzhou 510275, P.R.China

Received Date: 2025-04-02
Rev Recd Date: 2025-06-10
Publish Date: 2026-02-01

Abstract

Abstract

Firstly, as an example, the unstructured finite difference method (UFDM) based on discrete equivalent equations was proposed, to enhance the geometric adaptability of the finite difference method. During the solution of the heat conduction equations with the finite difference method in the curvilinear coordinate system, the coordinate transformation will lead to geometric-induced errors. This phenomenon was illustrated by means of the central difference scheme to solve the temperature field equation. Based on the precision definition of the truncation errors of the difference scheme, the geometry-induced error was theoretically demonstrated to inevitably leads to a reduction in order. Secondly, a linear preservation assessment model was built to verify the 1st-order accuracy, and the difference scheme was proved to be difficult to guarantee the 1st-order accuracy of the assessment model on non-uniform grids. On this basis, a linear preservation algorithm based on gradient reconstruction was proposed. Numerical calculations show that, for structured grids with any shape to calculate linearly distributed temperature fields, numerical solutions with errors at the machine precision level of 0 can be obtained. This study provides a theoretical and practical foundation for developing fully automatic temperature field calculation software.
- finite difference method,
- heat conduction equation,
- body-fitted coordinate system,
- accuracy assessment,
- verification and validation

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References(22)

References

[1]	PESKIN C S. Flow patterns around heart valves: a numerical method[J]. Journal of Computational Physics, 1972, 10(2): 252-271. doi: 10.1016/0021-9991(72)90065-4
[2]	ZHAO X, CHEN Z, YANG L, et al. Efficient boundary condition-enforced immersed boundary method for incompressible flows with moving boundaries[J]. Journal of Computational Physics, 2021, 441: 110425.
[3]	MITTAL R, SEO J H. Origin and evolution of immersed boundary methods in computational fluid dynamics[J]. Physical Review Fluids, 2023, 8(10): 100501.
[4]	刘君, 陈洁, 韩芳. 基于离散等价方程的非结构网格有限差分法[J]. 航空学报, 2020, 41(1): 123248. LIU Jun, CHEN Jie, HAN Fang. Finite difference method for unstructured grid based on discrete equivalent equation[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(1): 123248. (in Chinese)
[5]	刘君, 魏雁昕, 陈洁. 基于非结构网格有限差分法的扎染算法[J]. 航空学报, 2021, 42(7): 124557. LIU Jun, WEI Yanxin, CHEN Jie. Tie-dye algorithm based on finite difference method for unstructured grid[J]. Acta Aeronautica et Astronautica Sinica, 2021, 42(7): 124557. (in Chinese)
[6]	CHAWNER J R, DANNENHOFFER J, TAYLOR N J. Geometry, mesh generation, and the CFD 2030 vision[C]// 46th AIAA Fluid Dynamics Conference. Washington DC, Reston, Virginia: AIAA, 2016: 3485.
[7]	HINDMAN R G. Generalized coordinate forms of governing fluid equations and associated geometrically induced errors[J]. AIAA Journal, 1982, 20(10): 1359-1367. doi: 10.2514/3.51196
[8]	STEGER J L. Implicit finite difference simulation of flow about arbitrary geometries with application to airfoils[C]// 10th Fluid and Plasmadynamics Conference. Albuquerque NM, Reston, Virginia: AIAA, 1977: 665.
[9]	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
[10]	DENG X, MIN Y, MAO M, 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
[11]	NONOMURA T, TERAKADO D, ABE Y, et al. A new technique for freestream preservation of finite-difference WENO on curvilinear grid[J]. Computers & Fluids, 2015, 107: 242-255.
[12]	ZHU Y, SUN Z, REN Y, et al. A numerical strategy for freestream preservation of the high order weighted essentially non-oscillatory schemes on stationary curvilinear grids[J]. Journal of Scientific Computing, 2017, 72(3): 1021-1048. doi: 10.1007/s10915-017-0387-x
[13]	刘君, 魏雁昕, 韩芳. 有限差分法的坐标变换诱导误差[J]. 航空学报, 2021, 42(6): 124397. 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)
[14]	刘君, 白晓征, 郭正. 非结构动网格计算方法: 及其在包含运动界面的流场模拟中的应用[M]. 长沙: 国防科技大学出版社, 2009. LIU Jun, BAI Xiaozheng, GUO Zheng. Unstructured moving grid computation methods and their applications in flow field simulations involving moving interfaces[M]. Changsha: National University of Defense Technology Press, 2009. (in Chinese)
[15]	陶文铨. 计算流体力学与传热学[M]. 北京: 中国建筑工业出版社, 1991. TAO Wenquan. Computational Fluid Dynamics and Heat Transfer[M]. Beijing: China Architecture & Building Press, 1991. (in Chinese)
[16]	陶文铨. 计算传热学的近代进展[M]. 北京: 科学出版社, 2000. TAO Wenquan. Modern Progress of Computational Heat Transfer[M]. Beijing: Science Press, 2000. (in Chinese)
[17]	张德良. 计算流体力学教程[M]. 北京: 高等教育出版社, 2010. ZHANG Deliang. Computational Fluid Mechanics Tutorial[M]. Beijing: Higher Education Press, 2010. (in Chinese)
[18]	阎超. 计算流体力学方法及应用[M]. 北京: 北京航空航天大学出版社, 2006. YAN Chao. Computational Fluid Dynamics Method and Its Application[M]. Beijing: Beijing University of Aeronautics & Astronautics Press, 2006. (in Chinese)
[19]	刘君, 韩芳, 夏冰. 有限差分法中几何守恒律的机理及算法[J]. 空气动力学学报, 2018, 36(6): 917-926. 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)
[20]	刘君, 韩芳. 有限差分法中的贴体坐标变换[J]. 气体物理, 2018, 3(5): 18-29. LIU Jun, HAN Fang. Body-fitted coordinate transformation for finite difference method[J]. Physics of Gases, 2018, 3(5): 18-29. (in Chinese)
[21]	HAN F, XU C, 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
[22]	GUAN T, CHEN J, LIU J, et al. A second-order-accuracy finite difference scheme for numerical computation on two-dimensional three-neighboring-node unstructured grids[J]. Physics of Fluids, 2024, 36(3): 036109. doi: 10.1063/5.0193667