Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T07:08:49.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Effect of Nonuniform Grids on High-Order Finite Difference Method

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  18 January 2017

Dan Xu ,
Xiaogang Deng ,
Yaming Chen ,
Guangxue Wang  and
Yidao Dong
Dan Xu*
Affiliation:
College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, China
Xiaogang Deng
Affiliation:
College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, China
Yaming Chen
Affiliation:
College of Science, National University of Defense Technology, Changsha, Hunan 410073, China
Guangxue Wang
Affiliation:
School of Physics, Sun Yat-sen University, Guangzhou, Guangdong 510006, China
Yidao Dong
Affiliation:
College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, China
*
*Corresponding author. Email:[email protected] (D. Xu)
Get access

Abstract

The finite difference (FD) method is popular in the computational fluid dynamics and widely used in various flow simulations. Most of the FD schemes are developed on the uniform Cartesian grids; however, the use of nonuniform or curvilinear grids is inevitable for adapting to the complex configurations and the coordinate transformation is usually adopted. Therefore the question that whether the characteristics of the numerical schemes evaluated on the uniform grids can be preserved on the nonuniform grids arises, which is seldom discussed. Based on the one-dimensional wave equation, this paper systematically studies the characteristics of the high-order FD schemes on nonuniform grids, including the order of accuracy, resolution characteristics and the numerical stability. Especially, the Fourier analysis involving the metrics is presented for the first time and the relation between the resolution of numerical schemes and the stretching ratio of grids is discussed. Analysis shows that for smooth varying grids, these characteristics can be generally preserved after the coordinate transformation. Numerical tests also validate our conclusions.

Keywords

Finite difference methodnonuniform gridscoordinate transformationFourier analysis

MSC classification

Secondary: 65N06: Finite difference methods 65N22: Solution of discretized equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 4 , August 2017 , pp. 1012 - 1034
Copyright
Copyright © Global-Science Press 2017 

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] Vinokur, M., An analysis of finite-difference and finite-volume formulations of conservation laws, J. Comput. Phys., 81 (1989), pp. 1–52.Google Scholar
[2] Garmann, D. J., Compact finite-differencing and filtering procedure applied to the incompressible navier-stokes equations, AIAA J., 51(9) (2013).Google Scholar
[3] Zhang, X. and Shu, C. W., Positivity-preserving high order finite difference WENO schemes for compressible Euler equations, J. Comput. Phys., 231 (2012), pp. 2245–2258.Google Scholar
[4] Marxen, O., Iaccarino, G. and Magin, T. E., Direct numerical simulations of hypersonic boundary-layer transition with finite-rate chemistry, J. Fluid Mech., 755 (2014), pp. 35–49.Google Scholar
[5] Rizzetta, D. P., Visbal, M. R. and Morgan, P. E., A high-order compact finite-difference scheme for large-eddy simulation of active flow control, Progress Aerospace Sci., 44 (2008), pp. 397–426.Google Scholar
[6] Mor-Yossef, Y., Unconditionally stable time marching scheme for reynolds stress models, J. Comput. Phys., 276 (2014), pp. 635–664.Google Scholar
[7] Ekaterinaris, J. A., High-order accurate, low numerical diffusion methods for aerodynamics, Progress Aerospace Sci., 41 (2005), pp. 192–300.Google Scholar
[8] Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103 (1992), pp. 16–42.Google Scholar
[9] Deng, X., Jiang, Y., Mao, M., Liu, H., Li, S. and Tu, G., A family of hybrid cell-edge and cell-node dissipative compact schemes satisfying geometric conservation law, Comput. Fluids, 116 (2015), pp. 29–45.Google Scholar
[10] Cockburn, B. and Shu, C. W., Nonlinearly stable copact schemes for shock calculations, SIAM J. Numer. Anal., 31(3) (1994), pp. 607–627.Google Scholar
[11] Deng, X. and Maekawa, H., Compact high-order accurate nonlinear schemes, J. Comput. Phys., 130 (1997), pp. 77–91.Google Scholar
[12] Ren, Y.-X., Liu, M. and Zhang, H., A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws, J. Comput. Phys., 192 (2003), pp. 365–386.Google Scholar
[13] Shen, Y. and Zha, G., Generalized finite compact difference scheme for shock/complex flowfield interaction, J. Comput. Phys., 230 (2011), pp. 4419–4436.Google Scholar
[14] Tam, C. K. W., Computational Aeroacoustics: A Wave Number Approach, Cambridge University Press, 2012.Google Scholar
[15] Zingg, D. W., Rango, S. D., Nemec, M. and Pulliam, T. H., Comparison of several spatial discretizations for the Navier–Stokes equations, J. Comput. Phys., 160 (2000), pp. 683–704.Google Scholar
[16] Tam, C. K. W. and Webb, J. C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys., 107 (1993), pp. 262–281.Google Scholar
[17] Pirozzoli, S., Performance analysis and optimization of finite-difference schemes for wave propagation problems, J. Comput. Phys., 222 (2007), pp. 809–831.CrossRefGoogle Scholar
[18] Gamet, L., Ducros, F., Nicoud, F. and Poinsot, T., Compact finite difference schemes on non-uniform meshes, application to direct numerical simulations of compressible flows, Int. J. Numer. Meth. Fluids, 29 (1999), pp. 159–191.3.0.CO;2-9>CrossRefGoogle Scholar
[19] Cheong, C. and Lee, S., Grid-optimized dispersion-relation-preserving schemes on general geometries for computational aeroacoustics, J. Comput. Phys., 174 (2001), pp. 248–276.Google Scholar
[20] Zhong, X. and Tatineni, M., High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition, J. Comput. Phys., 190 (2003), pp. 419–458.Google Scholar
[21] Shukla, R. K. and Zhong, X., Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation, J. Comput. Phys., 204 (2005), pp. 404–429.Google Scholar
[22] Deng, X., Mao, M., Tu, G., Liu, H. and Zhang, H., Geometric conservation law and application to high-order finite difference schemes with stationary grids, J. Comput. Phys., 230 (2011), pp. 1100–1115.CrossRefGoogle Scholar
[23] Deng, X., Min, Y., Mao, M., Liu, H., Tu, G. and Zhang, H., Further study on geometric conservation law and application to high-order finite difference schemes with stationary grids, J. Comput. Phys., 239 (2013), pp. 90–111.Google Scholar
[24] Visbal, R. and Gaitonde, D., On the use of higher-order finite-difference schemes on curvilinear and deforming meshes, J. Comput. Phys., 181 (2002), pp. 155–185.Google Scholar
[25] Chung, Y. M. and Tucker, P. G., Accuracy of higher-order finite difference schemes on nonuniform grids, AIAA J., 41(8) (2003).CrossRefGoogle Scholar
[26] Whitham, G. B., Linear and Nonlinear Waves, John Wiley & Sons, 1974.CrossRefGoogle Scholar
[27] Deng, X. and Zhang, H., Developing high-order weighted compact nonlinear schemes, J. Comput. Phys., 165 (2000), pp. 22–44.Google Scholar
[28] Vinokur, M. and Yee, H., Extension of efficient low dissipation high-order schemes for 3D curvilinear moving grids, Front. Comput. Fluid Dyn., (2002), pp. 129–164.Google Scholar
[29] Nonomura, T., Iizuka, N. and Fujii, K., Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids, Comput. Fluids, 39 (2010), pp. 197–214.Google Scholar
[30] Vichnevetsky, R. and Bowles, J. B., Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM Philadelphia, 1982.Google Scholar
[31] Liu, X., Deng, X. and Mao, M., High-order behaviors of weighted compact fifth-order nonlinear schemes, AIAA J., 45(8) (2007).Google Scholar
[32] Carpenter, M. H., Gottlieb, D. and Abarbanel, S., The stablity of numerical boundary treatments for compact high-order finite-difference schemes, J. Comput. Phys., 108 (1993), pp. 272–295.Google Scholar
[33] Kosloff, D. and Tal-Ezer, H., A modified Chebyshev pseudo-spectral method with an time step restriction, J. Comput. Phys., 104 (1993), pp. 457–469.Google Scholar
[34] Liu, H., Deng, X., Mao, M. and Wang, G., High order nonlinear schemes for viscous terms and the application to complex configuration problems, AIAA, (2011), 2011-369.Google Scholar
[35] Casper, J., Shu, C. W. and Atkins, H., Comparision of two formulations for high-order accurate essentially nonoscillatory schemes, AIAA J., 32(10) (1994).Google Scholar
[36] Atkins, H. and Casper, J., Nonreflective boundary conditions for high-order methods, AIAA J., 32(3) (1994).Google Scholar
[37] Roberts, G. O., Computational meshes for the boundary problems, in: Second International Conference Numerical Methods Fluid Dynamics, Lecture Notes in Physics (Springer-Verlag, New York, 1971).Google Scholar
[38] Wang, Z. J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H. T., Kroll, N., May, G., Persson, P. O., Van Leer, B. and Visbal, M., High-order CFD methods: current status and perspective, Int. J. Numer. Meth. Fluids, 72 (2013), pp. 811–845.Google Scholar
[39] Abgrall, R. and Santis, D. D., Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier-Stokes equations, J. Comput. Phys., 283 (2015), pp. 329–359.Google Scholar
[40] Kroll, N., Adigma–a European project on the development of adaptive higher-order variational methods for aerospace applications, AIAA, (2009), 2009-176.Google Scholar
[41] Leicht, T. and Hartmann, R., Error estimation and anisotropic mesh refinement for 3D laminar aerodynamic flow simulations, J. Comput. Phys., 229 (2010), pp. 7344–7360.CrossRefGoogle Scholar