有限差分法的坐标变换诱导误差

doi:10.7527/S1000-6893.2020.24397

Abstract

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.

Key words: finite difference method, geometric conservation law, discrete equivalence equation, geometrically induced errors, coordinate transformation

CLC Number:

V211.3

LIU Jun, WEI Yanxin, HAN Fang. Coordinate transformation induced errors of finite difference method[J]. ACTA AERONAUTICAET ASTRONAUTICA SINICA, 2021, 42(6): 124397-124397.

References

[1] TRULIO J G, TRIGGER K R. Numerical solution of the one-dimensional Lagrangian hydrodynamic equations:UCRL-6267[R]. Oak Ridge:Office of Scientific and Technical Information (OSTI), 1961.
[2] STEGER J. Implicit finite difference simulation of flow about arbitrary geometries with application to airfoils[C]//10th Fluid and Plasmadynamics Conference. Reston:AIAA, 1977:665.
[3] STEGER J L. Implicit finite-difference simulation of flow about arbitrary two-dimensional geometries[J]. AIAA Journal, 1978, 16(7):679-686.
[4] PULLIAM T, STEGER J. On implicit finite-difference simulations of three-dimensional flow[C]//16th Aerospace Sciences Meeting. Reston:AIAA, 1978:2514.
[5] HINDMAN R. Geometrically induced errors and their relationship to the form of the governing equations and the treatment of generalized mappings[C]//5th Computational Fluid Dynamics Conference. Reston:AIAA, 1981:81-1008.
[6] HINDMAN R G. Generalized coordinate forms of governing fluid equations and associated geometrically induced errors[J]. AIAA Journal, 1982, 20(10):1359-1367.
[7] VIVIAND H, GHAZZI W. Numerical solution of the compressible Navier-Stokes equations at high Reynolds numbers with applications to the blunt body problem[C]//Proceedings of the 5th International Conference on Numerical Methods in Fluid Dynamics. Enschede:Twente University, 1976:434-439.
[8] THOMAS P, LOMBARD C. The Geometric Conservation Law-A link between finite-difference and finite-volume methods of flow computation on moving grids[C]//11th Fluid and Plasma Dynamics Conference. Reston:AIAA, 1978:1208.
[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.
[10] GAITONDE D, VISBAL M. Further development of a Navier-Stokes solution procedure based on higher-order formulas[C]//37th Aerospace Sciences Meeting and Exhibit. Reston:AIAA, 1999:557.
[11] DENG X G, MAO M L, TU G H, et al. Geometric conservation law and applications to high-order finite difference schemes with stationary grids[J]. Journal of Computational Physics, 2011, 230(4):1100-1115.
[12] DENG X G, MIN Y B, MAO M L, 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.
[13] 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.
[14] ZHU Y J, SUN Z S, REN Y X, 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.
[15] 朱志斌, 杨武兵, 禹旻. 满足几何守恒律的WENO格式及其应用[J]. 计算力学学报, 2017, 34(6):779-784. ZHU Z B, YANG W B, YU M. A WENO scheme with geometric conservation law and its application[J]. Chinese Journal of Computational Mechanics, 2017, 34(6):779-784(in Chinese).
[16] 闵耀兵. 高阶精度有限差分方法几何守恒律研究[D]. 绵阳:中国空气动力研究与发展中心, 2015. MIN Y B. The studies on geometric conservation law for high order finite difference method[D]. Mianyang:China Aerodynamics Research and Development Center, 2015(in Chinese).
[17] 张来平. 计算流体力学网格生成技术[M]. 北京:科学出版社, 2017:4. ZHANG L P. Mesh generation techniques in computational fluid dynamics[M]. Beijing:Science Press, 2017:4(in Chinese).
[18] 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 J, HAN F, XIA B. Mechanism and algorithm for geometric conservation law in finite difference method[J]. Acta Aerodynamica Sinica, 2018, 36(6):917-926(in Chinese).
[19] 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 J, HAN F. Body-fitted coordinate transformation for finite difference method[J]. Physics of Gases, 2018, 3(5):18-29(in Chinese).
[20] 刘君, 韩芳. 有关有限差分高精度格式两个应用问题的讨论[J]. 空气动力学学报, 2020, 38(2):244-253. LIU J, HAN F. Discussions on two problems in applications of high-order finite difference schemes[J]. Acta Aerodynamica Sinica, 2020, 38(2):244-253(in Chinese).
[21] 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.

Coordinate transformation induced errors of finite difference method

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