A survey of geometry conservation law for high-order finite difference method

MAO Meiliang; JIANG Yi; MIN Yaobing; ZHU Huajun; DENG Xiaogang

doi:10.7638/kqdlxxb-2020.0111

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A survey of geometry conservation law for high-order finite difference method

更新时间：2026-05-28

- A survey of geometry conservation law for high-order finite difference method
- Vol. 39, Issue 1, Pages: 157-167(2021)
- 作者机构：
  
  1. 中国空气动力研究与发展中心,绵阳,621000
  2. 军事科学院系统工程研究院,北京,100082
  3. 军事科学院,北京,100091
- 作者简介：
- 基金信息：
- DOI：10.7638/kqdlxxb-2020.0111
  CLC： V211.3
- Published：2021
- 稿件说明：
移动端阅览
毛枚良, 姜屹, 闵耀兵, 朱华君, 邓小刚. 高阶精度有限差分方法几何守恒律研究进展[J]. 空气动力学学报, 2021, 39(1): 157-167.

MAO Meiliang, JIANG Yi, MIN Yaobing, et al. A survey of geometry conservation law for high-order finite difference method[J]. 2021, 39(1): 157-167.
毛枚良, 姜屹, 闵耀兵, 朱华君, 邓小刚. 高阶精度有限差分方法几何守恒律研究进展[J]. 空气动力学学报, 2021, 39(1): 157-167. DOI： 10.7638/kqdlxxb-2020.0111.

MAO Meiliang, JIANG Yi, MIN Yaobing, et al. A survey of geometry conservation law for high-order finite difference method[J]. 2021, 39(1): 157-167. DOI： 10.7638/kqdlxxb-2020.0111.

摘要

针对高阶精度有限差分格式的几何守恒律问题，系统梳理了国内外离散几何守恒律问题方面的研究工作，以有限差分格式离散后的自由流保持问题为切入点，综述了近年来课题组在有限差分离散几何守恒律方面的研究工作，包括守恒网格导数算法（CMM）、对称守恒网格导数算法（SCMM）等，并通过若干典型算例验证了几何守恒律的满足对高阶精度有限差分方法数值模拟能力的提升。通过对有限差分离散几何守恒律问题研究工作的系统梳理，总结相关认识如下：1）直接基于网格变换导数的传统计算形式采用有限差分离散不能满足几何守恒律，需采用网格变换导数的守恒计算形式同时还需满足CMM条件；2）SCMM条件是满足几何守恒律的充分条件，且网格变换导数和雅克比均需采用其对称守恒计算形式，具有唯一性；3）自由流保持只是满足几何守恒律的一种表现形式；4）几何守恒律的满足能够有效提升高阶精度有限差分格式的数值模拟能力。

Abstract

Focusing on the geometry conservation law (GCL) for high-order finite difference schemes

the domestic and overseas studies regarding discretized geometric conservation law are reviewed systematically. Beginning with the freestream preservation phenomenon simulated by finite difference schemes

our recent studies about the geometric conservation law are summarized

including the conservative metric method (CMM) and the symmetrical conservative metric method (SCMM). Moreover

it is confirmed by several typical tests that the numerical simulations of high-order finite difference methods can be improved by satisfying the geometric conservation law. The systematic review about the geometric conservation law leads to the following conclusions: 1) The geometric conservation law cannot be satisfied by finite difference methods with the discretized metrics based on the traditional computational form. The conservative computational form of metrics together with the CMM condition should be adopted to satisfy the geometric conservation law; 2) The SCMM is a sufficient condition to satisfy the geometric conservation law

with an additional condition that the metrics and Jacobian must be discretized on the basis of the symmetrical conservative form

which are both unique; 3) The freestream preservation behavior is merely a representation of the satisfaction of the geometric conservation law; 4) The satisfaction of the geometric conservation law can effectively enhance the ability of numerical simulations with high-order finite difference schemes.

关键词

Keywords

references

CARUELLE B, DUCROS F. Detached-eddy simulations of attached and detached boundary layers[J]. International Journal of Computational Fluid Dynamics, 2003, 17: 433-451.

PHILIPPE R SPALART. Detached-eddy simulation[J]. Annual Review of Fluid Mechanics, 2009, 41: 181-202.

UGOPIOMELLI, ELIAS BALARAS. Wall-layer models for large-eddy simulations[J]. Annual Review of Fluid Mechanics, 2002, 34: 349-374.

PIERRE SAGAUT. Large eddy simulation for incompressible flows-An introduction[M]. Third edition, Springer, 2006.

PARVIZ MOIN, KRISHNAN MAHESH. Direct numerical simulation: A tool in turbulence research[J]. Annual Review of Fluid Mechanics, 1998, 30: 539-578.

TAKASHI ISHIHARA, TOSHIYUKIGOTOH, YUKIO KANEDA. Study of high-Reynolds number isotropic turbulence by direct numerical simulation[J]. Annual Review of Fluid Mechanics, 2009, 41: 165-180.

SHEN J, TANG T, WANG L L. Spectral methods: Algorithms, analysis and applications[M]. Springer, 2011.

BERNARDO COCKBURN, SHU C W. The local discontinuous Galerkin method for time-dependent convection-diffusion systems[J]. SIAM Journal on Numerical Analysis, 1998, 35: 2440-2463.

BERNARDO COCKBURN, SHU C W. Runge-Kutta discontinuous Galerkin methods for convection-dominated problems[J]. Journal of Scientific Computing, 2001, 16(3): 173-261.

YEN LIU, MARCEL VINOKUR, WANG Z J. Spectral difference method for unstructured grids I: Basic formulation[J]. Journal of Computational Physics, 2006, 216: 780-801.

WANG Z J, LIU Y, GEORG MAY, et al. Spectral difference method for unstructured grids Ⅱ: Extension to the Euler equations[J]. Journal of Scientific Computing, 2007, 32: 45-71.

WANG Z J. Spectral (finite) volume method for conservation laws on unstructured grids: Basic formulation[J]. Journal of Computational Physics, 2002, 178: 210-251.

WANG Z J, LIU Y. Spectral (finite) volume method for conservation laws on unstructured grids Ⅱ: Extension to two-dimensional scalar equation[J]. Journal of Computational Physics, 2002, 179: 665-697.

DINSHAW S BALSARA, SHU C W. Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy[J]. Journal of Computational Physics, 2000, 160: 405-452.

DONG Y D, DENG X G, XU D, et al. Reevaluation of high-order finite difference and finite volume algorithms with freestream preservation satisfied[J]. Computers and Fluids, 2017, 156: 343-352.

JAY CASPER, SHU C W, ATKINS H. Comparison of two formulations for high-order accurate essentially non-oscillatory schemes[J]. American Institute of Aeronautics and Astronautics Journal, 1994, 32(10): 1970-1977.

XU C Y, CHEN L W, LU X Y. Large-eddy simulation of the compressible flow past a wavy cylinder[J]. Journal of Fluid Mechanics, 2010, 665: 238-273.

ERIC LAMBALLAIS. Direct numerical simulation of a turbulent flow in a rotating channel with a sudden expansion[J]. Journal of Fluid Mechanics, 2014, 745: 99-131.

HERVÉ GUILLARD, CHARBEL FARHAT. On the significance of the geometric conservation law for flow computations on moving meshes[J]. Computer Methods in Applied Mechanics and Engineering, 2000, 190: 1467-1482.

DANIELE BOFFI, LUCIA GASTALDI. Stability and geometric conservation laws for ALE formulations[J]. Computer Methods in Applied Mechanics and Engineering, 2004, 193: 4717-4739.

DIMITRI J MAVRIPLIS, YANG Z. Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes[J]. Journal of Computational Physics, 2006, 213: 557-573.

DIMITRI J MAVRIPLIS, CRISTIAN R NASTASE. On the geometric conservation law for high-order discontinuous Galerkin discretizations on dynamically deforming meshes[J]. Journal of Computational Physics, 2011, 230: 4285-4300.

THOMAS P D, LOMBARD C K. The Geometric conservation law-a link between finite difference and finite volume methods of flow computation on moving grids: AIAA-78-1208[R]. American Institute of Aeronautics and Astronautics, 1978.

THOMAS P D, LOMBARD C K. Geometric conservation law and its application to flow computations on moving grids[J]. American Institute of Aeronautics and Astronautics Journal, 1979, 17(10): 1030-1037.

STEGER J L. Implicit finite difference simulation of flow about arbitrary geometrics with application to airfoils: AIAA-77-665[R]. American Institute of Aeronautics and Astronautics, 1977.

JOSEPH L STEGER. Implicit finite difference simulation of flow about arbitrary two dimensional geometries[J]. American Institute of Aeronautics and Astronautics Journal, 1978, 16(7): 679-686.

ZHANG H, REGGIO M, TRÉPANIER J Y, et al. Discrete form of the GCL for moving meshes and its implementation in CFD schemes[J]. Computers and Fluids, 1993, 22(1): 9-23.

YOSHIAKI ABE, NOBUYUKⅢZUKA, TAKU NONOMURA, et al. Conservative metric evaluation for high-order finite difference schemes with the GCL identities on moving and deforming grids[J]. Journal of Computational Physics, 2013, 232: 14-21.

YOSHIAKI ABE, TAKU NONOMURA, NOBUYUKI ⅡZUKA, et al. Geometric interpretations and spatial symmetry property of metrics in the conservative form for high-order finite-difference schemes on moving and deforming grids[J]. Journal of Computational Physics, 2014, 260: 163-203.

WANG Z J, YANG H Q. Unsteady flow simulation using a zonal multi-grid approach with moving boundaries: AIAA-94-0057[R]. American Institute of Aeronautics and Astronautics, 1994.

ZHANG L P, WANG Z J. A block LU-SGS implicit dual time-stepping algorithm for hybrid dynamic meshes[J]. Computers and Fluids, 2004, 33: 891-916.

CHANG X H, MA R, ZHANG L P, et al. Further study on the geometric conservation law for finite volume method on dynamic unstructured mesh[J]. Computers and Fluids, 2015, 120: 98-110.

RICHARD G HINDMAN. Geometrically induced errors and their relationship to the form of the governing equations and the treatment of generalized mappings: AIAA-81-1008[R]. American Institute of Aeronautics and Astronautics, 1981.

RICHARD G HINDMAN. Generalized coordinate forms of governing fluid equations and associated geometrically induced errors[J]. American Institute of Aeronautics and Astronautics Journal, 1982, 20(10): 1359-1367.

HENRIVIVIAND, WALID GHAZZI. Numerical solution of the compressible Navier-Stokes equations at high Reynolds numbers with applications to the blunt body problem[J]. Lecture Notes in Physics, 1976, 59: 434-439.

PULLIAM T H, STEGER J L. On implicit finite-difference simulations of three dimensional flow: AIAA-78-10[R]. American Institute of Aeronautics and Astronautics, 1978.

THOMAS H PULLIAM, JOSEPH L STEGER. Implicit finite-difference simulation of three dimensional compressible flow[J]. American Institute of Aeronautics and Astronautics Journal, 1980, 18(2): 159-167.

MIGUEL R VISBAL, DATTA V GAITONDE. On the use of higher-order finite-difference schemes on curvilinear and deforming meshes[J]. Journal of Computational Physics, 2002, 181: 155-185.

TAKU NONOMURA, NOBUYUKI ⅡZUKA, KOZO FUJⅡ. Free-stream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids[J]. Computers and Fluids, 2010, 39: 197-214.

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: 1100-1115.

DENG X G, ZHANG H X. Developing high-order weighted compact nonlinear schemes[J]. Journal of Computational Physics, 2000, 165: 22-44.

邓小刚. 高阶精度耗散加权紧致非线性格式[J]. 中国科学(A辑), 2001, 31(12): 1104-1117.

TAKU NONOMURA, KOZO FUJⅡ. Effects of difference scheme type in high-order weighted compact nonlinear schemes[J]. Journal of Computational Physics, 2009, 228: 3533-3539.

JIANG G S, SHU C W. Efficient implementation of weighted ENO schemes[J]. Journal of Computational Physics, 1996, 126: 202-228.

SHU C W. High order weighted essentially non-oscillatory schemes for convection dominated problems[J]. SIAM Review, 2009, 51: 82-126.

TAKU NONOMURA, DAIKI TERAKADO, YOSHIAKI ABE, et al. A new technique for freestream preservation of finite difference WENO on curvilinear grid[J]. Computers and Fluids, 2015, 107: 242-255.

JIANG Y, SHU C W, ZHANG M P. Free-stream preserving finite difference schemes on curvilinear meshes[R]. Brown University, Scientific Computing Group Report 2013-10, 2013.

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.

DENG X G, ZHU H J, MIN Y B, et al. Symmetric conservative metric method: A link between high order finite-difference and finite-volume schemes for flow computations around complex geometries[C]//Eighth International Conference on Computational Fluid Dynamics, Chengdu, China, ICCFD8-2014-0005, 2014.

DENG X G, ZHU H J, Min Y B, et al. High-order finite difference schemes based on symmetric conservative metric method: Decomposition, geometric meaning and connection with finite volume schemes[J]. Advances in Applied Mathematics and Mechanics, 2020, 12(2): 436-479.

DENG X G, MAO M L, TU G H, et al. Extending weighted compact nonlinear schemes to complex grids with characteristic-based interface conditions[J]. American Institute of Aeronautics and Astronautics Journal, 2010, 48(12): 2840-2851.

KIM J W, LEE D J. Generalized characteristic boundary conditions for computational aeroacoustics, Part 2[J]. American Institute of Aeronautics and Astronautics Journal, 2004, 42(1): 47-55.

MAO M L, ZHU H J, DENG X G, et al. Effect of geometric conservation law on improving spatial accuracy for finite difference schemes on two-dimensional nonsmooth grids[J]. Communications in Computational Physics, 2015, 18(3): 673-706.

MIN Y B, MA Y K, LI S. Accuracy analysis of numerical error with statistical forms in CFD[J]. Acta Aeronautica et Astronautica Sinica, 2020, 41(4): 123554. (in Chinese)

DENG X G, YI Y, MAO M L, et al. A family of hybrid cell-edge and cell-node dissipative compact schemes satisfying geometric conservation law[J]. Computers and Fluids, 2015, 116: 29-45.

JIANG Y, DENG X G, MAO M L, et al, Progress of aeroacoustic simulation method based on HDCS-E8T7 scheme[J]. Acta Aerodynamica Sinica, 2014, 32(5): 559-574. (in Chinese)

YI Y, MAO M L, DENG X G, et al. Extending seventh-order dissipative compact scheme satisfying geometric conservation law to large eddy simulation on curvilinear grids[J]. Advances in Applied Mathematics and Mechanics, 2015, 7(4): 407-429.

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