有限差分法计算热传导方程的几何诱导误差

刘君; 刘光英; 徐春光

doi:10.21656/1000-0887.460061

有限差分法计算热传导方程的几何诱导误差

doi: 10.21656/1000-0887.460061

1.
宁波大学机械工程与力学学院, 浙江宁波 315211
2.
中山大学航空航天学院, 广州 510275

详细信息

通讯作者:
刘君(1965—)，男，教授，博士，博士生导师(通信作者. E-mail: liujun65@dlut.edu.cn)

中图分类号: O35
计量
- 文章访问数: 12
- HTML全文浏览量: 13
- PDF下载量: 1
- 被引次数: 0
出版历程
- 收稿日期: 2025-04-02
- 修回日期: 2025-06-10
- 刊出日期: 2026-02-01

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

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

摘要

摘要: 首先举例说明，基于离散等价方程提出了非结构网格有限差分法(UFDM)，提高了有限差分法几何适应性. 在有限差分法求解曲线坐标系下热传导方程时，坐标变换导致出现几何诱导误差. 这一现象通过采用中心格式求解温度场方程给出说明. 根据差分格式截断误差的精度定义，理论上论证了几何诱导误差导致降阶的必然性. 其次提出了验证一阶精度的保线性考核模型，据此得出了差分格式在非均匀网格难以保证考核模型一阶精度的结论. 在此基础上，提出了基于梯度重构的保线性算法. 数值计算表明，对任意形状结构网格计算线性分布温度场，均可得到误差为机器精度0量级的数值解，为开发全自动温度场计算软件提供了理论与实践基础.
- 有限差分法 /
- 热传导方程 /
- 贴体坐标系 /
- 精度评估 /
- 验证与确认
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

HTML全文

图 1 非水密几何的差分示意

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 1. Schematic of differential for non-water-tight geometry

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图 2 基于点云的扎染算法全自动流场仿真演示

Figure 2. Demonstration of fully automatic flow field simulation based on point cloud for the tie-dye algorithm

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图 3 均匀网格误差

Figure 3. Uniform grid errors

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图 4 挪动一列点后的非均匀网格

Figure 4. The non-uniform grid after shifting a column of points

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图 5 非均匀网格误差

Figure 5. Errors of the non-uniform grid

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图 6 网格扰动系数引起的误差变化

Figure 6. Error variations caused by grid disturbance coefficients

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图 7 圆柱坐标网格

Figure 7. The cylindrical coordinate grid

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图 8 圆柱网格误差

Figure 8. Errors of the cylindrical grid

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图 9 网格加密变化误差收敛阶

Figure 9. Convergence orders of error variations due to the grid refinement

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图 10 非共面四边形网格示意图和误差云图

Figure 10. Schematic of the non-coplanar quadrilateral grid and the error cloud map

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图 11 柱坐标消减误差(81×81)

Figure 11. Error reductions in the cylindrical coordinates (81×81)

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图 12 柱坐标消减误差(21×21)

Figure 12. Error reductions in the cylindrical coordinates (21×21)

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图 13 波浪网格误差

Figure 13. Errors of the wavy grid

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图 14 随机扰动网格误差

Figure 14. Errors of the randomly disturbed grid

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表 1 网格加密引起的误差

Table 1. Errors caused by the grid refinement

Δx	ε(L²)	ε(L^∞)
0.05	6.29E-7	2.60E-5
0.025	8.26E-8	6.53E-6
0.012 5	1.06E-8	1.63E-6
0.006 25	1.34E-9	4.09E-7

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