### 有限差分法的坐标变换诱导误差

1. 大连理工大学 航空航天学院, 大连 116024
• 收稿日期:2020-06-11 修回日期:2020-09-18 出版日期:2021-06-15 发布日期:1900-01-01
• 通讯作者: 刘君 E-mail:liujun65@dlut.edu.cn
• 基金资助:
国家重点研发计划（2018YFB0204404）；国家自然科学基金（11872144）

### Coordinate transformation induced errors of finite difference method

LIU Jun, WEI Yanxin, HAN Fang

1. School of Aeronautics and Astronautics, Dalian University of Technology, Dalian 116024, China
• Received:2020-06-11 Revised:2020-09-18 Online:2021-06-15 Published:1900-01-01
• Supported by:
National Key Research & Development Program of China (2018YFB0204404); National Natural Science Foundation of China (11872144)

Abstract: Coordinate transformation is required when the finite difference method is applied to the mesh with complex geometries, and the errors induced by the coordinate transformation are often introduced in this process. These errors are proved to be inevitable in the uniform flow field calculation with uniform grids in cylindrical coordinate systems, even if the transformation function of the physical coordinates to the calculated coordinates is continuously derivable, or the coordinate transformation coefficients in the calculation process are calculated by the accurate analytical formula, or the grid is completely orthogonal and fully smooth. Theoretical analysis shows the mechanism of the coordinate transformation induced errors:when the conservative Euler equation is transformed from the Cartesian coordinate system to the body fitted coordinate system, a source term is added. Currently, scholars usually use the geometric conservation law to construct a method based on coordinate transformation coefficients, which are matched with the format of the finite difference, to eliminate the source term. In this work, we introduce a new algorithm that processes the direct discretization from the discrete equivalent functions including the source term. Based on the above new algorithm, error analysis is carried out for the reconstruction process of MUSCL format under non-equidistant grid conditions. Theoretical derivation shows that the influence coefficient of the non-equidistant interpolation formula needs to be considered in reconstruction, only when the variables are transformed into the computational space for MUSCL reconstruction can the interpolation accuracy be guaranteed under uniform grid. Our theoretical analysis and numerical experiments have proven that this algorithm will not introduce coordinate transformation errors to the uniform flow field calculations.