Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-24T09:17:30.768Z Has data issue: false hasContentIssue false
  • English
  • Français

Effect of Geometric Conservation Law on Improving Spatial Accuracy for Finite Difference Schemes on Two-Dimensional Nonsmooth Grids

Part of: Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical approximation and computational geometry

Published online by Cambridge University Press:  14 September 2015

Meiliang Mao ,
Huajun Zhu ,
Xiaogang Deng ,
Yaobing Min  and
Huayong Liu
Meiliang Mao*
Affiliation:
State Key Laboratory of Aerodynamics, CARDC, Mianyang, 621000, P.R. China Computational Aerodynamics Institute, CARDC, Mianyang, 621000, P.R. China
Huajun Zhu
Affiliation:
State Key Laboratory of Aerodynamics, CARDC, Mianyang, 621000, P.R. China
Xiaogang Deng
Affiliation:
National University of Defense Technology, Changsha, Hunan, 410073, P.R. China
Yaobing Min
Affiliation:
State Key Laboratory of Aerodynamics, CARDC, Mianyang, 621000, P.R. China
Huayong Liu
Affiliation:
State Key Laboratory of Aerodynamics, CARDC, Mianyang, 621000, P.R. China
*
*Corresponding author. Email addresses: [email protected] (M. Mao), [email protected] (H. Zhu), [email protected] (X. Deng), [email protected] (Y. Min), [email protected] (H. Liu)
Get access

Abstract

It is well known that grid discontinuities have significant impact on the performance of finite difference schemes (FDSs). The geometric conservation law (GCL) is very important for FDSs on reducing numerical oscillations and ensuring free-stream preservation in curvilinear coordinate system. It is not quite clear how GCL works in finite difference method and how GCL errors affect spatial discretization errors especially in nonsmooth grids. In this paper, a method is developed to analyze the impact of grid discontinuities on the GCL errors and spatial discretization errors. A violation of GCL cause GCL errors which depend on grid smoothness, grid metrics method and finite difference operators. As a result there are more source terms in spatial discretization errors. The analysis shows that the spatial discretization accuracy on non-sufficiently smooth grids is determined by the discontinuity order of grids and can approach one higher order by following GCL. For sufficiently smooth grids, the spatial discretization accuracy is determined by the order of FDSs and FDSs satisfying the GCL can obtain smaller spatial discretization errors. Numerical tests have been done by the second-order and fourth-order FDSs to verify the theoretical results.

Keywords

Geometric conservation lawfinite difference schemespatial discretization errornonsmooth gridshigh-order accuracy

MSC classification

Secondary: 35L04: Initial-boundary value problems for first-order hyperbolic equations 65D25: Numerical differentiation 65D18: Computer graphics, image analysis, and computational geometry 65M06: Finite difference methods 65M15: Error bounds
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 3 , September 2015 , pp. 673 - 706
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Rango, S. D. and Zingg, D.W.. Aerodynamic computations using a higher-order algorithm. AIAA paper 99–0167, 1999.Google Scholar
[2]Rango, S. D. and Zingg, D. W.. Further investigation of a higher-order algorithm for aerodynamic computations. AIAA paper 2000–0823, 2000.Google Scholar
[3] J.A. Ekaterinaris. High-order accurate, low numerical diffusion methods for aerodynamics. Progress in Aerospace Sciences, 41:192–300, 2005.Google Scholar
[4]Wang, Z.J., Fidkowski, Krzysztof, and etal. High-order cfd methods: current status and perspective. Int. J. Numer. Meth. Fluids, 00:1–42, 2012.Google Scholar
[5]Visbal, M.R. and Gaitonde, D.V.. On the use of higher-order finite-difference schemes on curvilinear and deforming meshes. J. Comput. Phys., 181:155–185, 2002.Google Scholar
[6]Casper, J., Shu, C.W., and Atkins, H.. Comparison of two formulations for high-order accurate essentially nonoscillatory schemes. AIAA Journal, 32:1970–1977, 1994.Google Scholar
[7]Castillo, J.E., Hyman, J.M., Shashkov, M.J., and Steinberg, S.. The sensitivity and accuracy of fourth order finite-difference schemes on nonuniform grids in one dimension. Computers Math. Applic., 30:41–55, 1995.Google Scholar
[8]Shu, C.W.. High-order finite difference and finite volume weno schemes and discontinuous galerkin methods for cfd. Int. J. Comput. Fluid Dynamics, 17:107–118, 2003.Google Scholar
[9]Pulliam, T.H. and Steger, J.L.. On implicit finite-difference simulations of three-dimensional flow. AIAA Paper 78–10, 1978.Google Scholar
[10]Thomas, P.D. and Lombard, C.K.. The geometric conservation law-a link between finite difference and finite volume methods of flow computation on moving grids. AIAA paper 78–1208, 1978.Google Scholar
[11]Thomas, P. D. and Lombard, C. K.. Geometric conservation law and its application to flow computations on moving grids. AIAA Journal, 17(10):1030–1037, 1979.CrossRefGoogle Scholar
[12]Zhang, H., Reggio, M., TršŠpanier, J.Y., and Camarero, R.. Discrete form of the gcl for moving meshes and its implementation in cfd schemes. Computers and Fluids, 22(1):9–23, 1993.Google Scholar
[13]Lesoinne, M. and Farhat, C.. Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Comput. Methods Appl. Mech. Eng., 134:71–90, 1996.Google Scholar
[14]Guillard, H. and Farhat, C.. On the significance of the geometric conservation law for flow computations on moving meshes. Comput. Methods Appl. Mech. Eng., 190:1467–1482, 2000.Google Scholar
[15]Geuzaine, P., Grandmont, C., and Farhat, C.. Design and analysis of ale schemes with provable second-order time-accuracy for inviscid and viscous flow simulations. J. Comput. Phys., 191 (2003), pp. ., 191:206–227, 2003.Google Scholar
[16]Mavriplis, D.J. and Yang, Z.. Construction of the discrete geometric conservation law for high-order time accurate simulations on dynamic meshes. J. Comput. Phys., 213 (2):557–573, 2006.Google Scholar
[17]Sitaraman, J. and Baeder, J.D.. Field velocity approach and geometric conservation law for unsteady flow simulations. AIAA Journal, 44(9):2084–2094, 2006.Google Scholar
[18]Etienne, S., Garon, A., and Pelletier, D.. Geometric conservation law and finite element methods for ale simulations of incompressible flow. AIAA Paper 2008–733, 2008.Google Scholar
[19]Mavriplis, D.J. and Nastase, C.R.. On the geometric conservation law for high-order discontinuous galerkin discretizations on dynamically deforming meshes. J. Comput. Phys., 230:4285–4300, 2011.Google Scholar
[20]Sjogreen, B., Yee, H. C., and Vinokur, M.. On high order finite-difference metric discretizations satisfying gcl on moving and deforming grids. Technical report, Lawrence Livermore National Laboratory, LLNL-TR-637397, 2013.CrossRefGoogle Scholar
[21]Farhat, C., Geuzainne, P., and Grandmont, C.. The discrete geometric conservation law and the nonlinear stability of ale schemes for the solution of flow problems on moving grids. J. Comput. Phys., 174:669–694, 2001.Google Scholar
[22]Formaggia, L. and Nobile, F.. Stability analysis of second-order time accurate schemes for ale-fem. Comput. Methods Appl. Mech. Eng., 193:4097–4116, 2004.Google Scholar
[23]Ou, K. and Jameson, A.. On the temporal and spatial accuracy of spectral difference method on moving deformable grids and the effect of geometry conservation law. AIAA Paper 2010–5032, 2010.Google Scholar
[24]Deng, X., Jiang, Y., Mao, M., Liu, H., and Tu, G.. Developing hybrid cell-edge and cell-node dissipative compact scheme for complex geometry flows. Science China, 56(10):2361–2369, 2013.Google Scholar
[25]Nonomura, T., Iizuka, N., and Fujii, K.. Freestream and vortex preservation properties of high-order weno and wcns on curvilinear grids. Computers and Fluids, 39:197–214, 2010.Google Scholar
[26]Deng, X. and Zhang, H.. Developing high-order weighted compact nonlinear schemes. J. Comput. Phys., 165:22–44, 2000.Google Scholar
[27]Deng, X.. High-order accurate dissipative weighted compact nonlinear schemes. Science in China (Series A) x, 45 (3):356–370, 2002.Google Scholar
[28]Deng, X., Liu, X., and Mao, M.. Advances in high-order accurate weighted compact nonlinear schemes. Advances in Mechanics, 37 (3):417–427, 2007.Google Scholar
[29]Jiang, G.S. and Shu, C.W.. Efficient implementation of weighted eno schemes. J. Comput. Phys., 126:202–228, 1996.Google Scholar
[30]Deng, X., Mao, M., Tu, G., Liu, H., and Zhang, H.. Geometric conservation law and applications to high-order finite difference schemes with stationary grids. J. Comput. Phys., 230:1100–1115, 2011.Google Scholar
[31]Nonomura, T., Terakado, D., Abe, Y., and Fujii, K.. A new technique for finite difference weno with geometric conservation law. AIAA Paper 2013–2569, 2013.Google Scholar
[32]Deng, X., Min, Y., Mao, M., Liu, H., Tu, G., and Zhang, H.. Further studies on geometric conservation law and applications to high-order finite difference schemes with stationary grids. J. Comput. Phys., 239:90–111, 2013.Google Scholar