差分格式应用于非均匀网格的精度保持算法

刘君; 姜金裕

doi:10.7638/kqdlxxb-2025.0023

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差分格式应用于非均匀网格的精度保持算法

研究论文 | 更新时间：2026-07-11

- 差分格式应用于非均匀网格的精度保持算法
- Accuracy-preserving algorithm for difference schemes applied to non-uniform grids
- 空气动力学学报 2026年44卷第6期页码：25-33
- 作者机构：
  
  大连理工大学航空航天学院,大连,116024
- 作者简介：
  
  刘君*，研究方向：计算流体力学、空气动力学．E-mail：liujun65@dlut.edu.cn
- 基金信息：
- DOI：10.7638/kqdlxxb-2025.0023
  中图分类号： V211.3
- 纸质出版：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

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