Advertisement

Applied Mathematics and Mechanics

, Volume 28, Issue 7, pp 943–953 | Cite as

Three-point explicit compact difference scheme with arbitrary order of accuracy and its application in CFD

  • Lin Jian-guo  (林建国)Email author
  • Xie Zhi-hua  (谢志华)
  • Zhou Jun-tao  (周俊陶)
Article
  • 163 Downloads

Abstract

Based on the successive iteration in the Taylor series expansion method, a three-point explicit compact difference scheme with arbitrary order of accuracy is derived in this paper. Numerical characteristics of the scheme are studied by the Fourier analysis. Unlike the conventional compact difference schemes which need to solve the equation to obtain the unknown derivatives in each node, the proposed scheme is explicit and can achieve arbitrary order of accuracy in space. Application examples for the convection-diffusion problem with a sharp front gradient and the typical lid-driven cavity flow are given. It is found that the proposed compact scheme is not only simple to implement and economical to use, but also is effective to simulate the convection-dominated problem and obtain high-order accurate solution in coarse grid systems.

Key words

arbitrary order of accuracy compact scheme three-point stencil explicit lid-driven cavity flow 

Chinese Library Classification

O241.82 X145 

2000 Mathematics Subject Classification

35Q30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Carpenter M H, Gottlieb D, Abarbanel S. The stability of numerical boundary treatments for compact high-order finite-difference schemes[J]. Journal of Computational Physics, 1993, 108(2):272–295.zbMATHCrossRefGoogle Scholar
  2. [2]
    Lele S K. Compact finite difference schemes with spectral-like resolution[J]. Journal of Computational Physics, 1992, 103(1):16–42.zbMATHCrossRefGoogle Scholar
  3. [3]
    Chu P C, Fan Chenwu. A three-point combined compact difference scheme[J]. Journal of Computational Physics, 1998, 140(2):370–399.zbMATHCrossRefGoogle Scholar
  4. [4]
    Mahesh K. A family of high order finite difference schemes with good spectral resolution[J]. Journal of Computational Physics, 1998, 145(1):332–358.zbMATHCrossRefGoogle Scholar
  5. [5]
    Hixon R. Prefactored Small-Stencil Compact Schemes[J]. Journal of Computational Physics, 2000, 165(2):522–541.zbMATHCrossRefGoogle Scholar
  6. [6]
    Tolstykh A I, Lipavskii M V. On performance of methods with third-and fifth-order compact upwind differencing[J]. Journal of Computational Physics, 1998, 140(2):205–232.zbMATHCrossRefGoogle Scholar
  7. [7]
    Ma Yanwen, Fu Dexun, Kobayashi N, et al. Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme[J]. International Journal for Numerical Methods in Fluids, 1999, 30(5):509–521.zbMATHCrossRefGoogle Scholar
  8. [8]
    Ma Yanwen, Fu Dexun. Analysis of super compact finite difference method and application to simulation of vortex-shock interaction[J]. International Journal for Numerical Methods in Fluids, 2001, 36(7):773–805.zbMATHCrossRefGoogle Scholar
  9. [9]
    Boersma B J. A staggered compact finite difference formulation for the compressible Navier-Stokes equations[J]. Journal of Computational Physics, 2005, 208(2):675–690.zbMATHCrossRefGoogle Scholar
  10. [10]
    Yuan Xiangjiang, Zhou Heng. Numerical schemes with high order of accuracy for the computation of shock waves[J]. Applied Mathematics and Mechanics (English Edition), 2000, 21(5):489–500.zbMATHCrossRefGoogle Scholar
  11. [11]
    Liu Ruxun, Wu Lingling. Small-stencil Padé schemes to solve nonlinear evolution equations[J]. Applied Mathematics and Mechanics (English Edition), 2005, 26(7):872–881.CrossRefGoogle Scholar
  12. [12]
    Fomberg B, Ghrist M. Spatial finite difference approximations for wave-type equation[J]. SIAM Journal on Numerical Analysis, 1999, 37(1):105–130.CrossRefGoogle Scholar
  13. [13]
    Lin Jianguo, Qiu Dahong. The Boussinesq type equation with the second order nonlinearity and dispersion[J]. Science in China, Series E, 1998, 28(6):567–573 (in Chinese).Google Scholar
  14. [14]
    Spotz W F. High order compact finite difference schemes for computational mechanics[D]. University of Texas at Austin, December 1995.Google Scholar
  15. [15]
    Kalita J C, Dalal D C, Dass A K. A class of higher order compact schemes for the unsteady two-dimensional convection diffusion equation with variable convection coefficients[J]. International Journal for Numerical Methods in Fluids, 2002, 38(12):1111–1131.zbMATHCrossRefGoogle Scholar
  16. [16]
    Ghia U, Ghia K N, Shin C T. High-Re solutions for imcompressible flow using the Navier-Stokes equation and a multigrid method[J]. Journal of Computational Physics, 1982, 48(3):387–411.zbMATHCrossRefGoogle Scholar

Copyright information

© Editorial Committee of Appl. Math. Mech. 2007

Authors and Affiliations

  • Lin Jian-guo  (林建国)
    • 1
    Email author
  • Xie Zhi-hua  (谢志华)
    • 1
  • Zhou Jun-tao  (周俊陶)
    • 1
  1. 1.College of Environmental Science and EngineeringDalian Maritime UniversityDalianP. R. China

Personalised recommendations