Acta Mechanica Sinica

, Volume 16, Issue 3, pp 223–239 | Cite as

Numerical solutions of incompressible Euler and Navier-Stokes equations by efficient discrete singular convolution method

  • D. C. Wan
  • G. W. Wei


An efficient discrete singular convolution (DSC) method is introduced to the numerical solutions of incompressible Euler and Navier-Stokes equations with periodic boundary conditions. Two numerical tests of two-dimensional Navier-Stokes equations with periodic boundary conditions and Euler equations for doubly periodic shear layer flows are carried out by using the DSC method for spatial derivatives and fourth-order Runge-Kutta method for time advancement, respectively. The computational results show that the DSC method is efficient and robust for solving the problems of incompressible flows, and has the potential of being extended to numerically solve much broader problems in fluid dynamics.

Key words

incompressible flows periodic boundary DSC method fourth-order Runge-Kutta method 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Forsythe GE, Wasow WR. Finite-Difference Methods for Partial Differential Equations. New York: Wiley, 1960zbMATHGoogle Scholar
  2. 2.
    Zienkiewicz OC. The Finite Element Method in Engineering Science. London: McGraw_Hill, 1971zbMATHGoogle Scholar
  3. 3.
    Cheung YK. Finite Strip Methods in Structural Analysis. Oxford: Pergamon Press, 1976Google Scholar
  4. 4.
    Patankar SV. Numerical Heat Transfer and Fluid Flow. New York: McGraw-Hill, 1980zbMATHGoogle Scholar
  5. 5.
    Harten A. High resolution schemes for hyperbolic conservation laws.J Comput Phys, 1983, 49: 357–393zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Harten A, Osher S. Uniformly high-order accurate nonoscillatory schemes I.SIAM J Numer Anal, 1987, 24: 279–309zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Hirsch R. Higher-order accurate difference solutions of a fluid mechanics problems by a compact differencing technique.J Comput Phys, 1975, 19: 90–109CrossRefGoogle Scholar
  8. 8.
    Andrei I Tolstykh. High accuracy non-centered compact difference schemes for fluid dynamics applications. In: Series on Advances in Mathematics for Applied Sciences, Volume 21. Singapore: World Scientific Publishing Co Pte Ltd, 1994Google Scholar
  9. 9.
    Weinan E, Liu JG. Essentially compact schemes for unsteady viscous incompressible flows.J Comput Phys, 1996, 126: 122–138MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gottlieb D, Hussaini MY, Orszag SA. Spectral methods for partial differential equations. In: Voigt RG, et al ed. SIAM, Philadelphia, 1984Google Scholar
  11. 11.
    Canuto C, Hussaini MY, Quarteroni A, Zang TA. Spectral Methods in Fluid Dynamics, Berlin: Springer-Verlag, 1988zbMATHGoogle Scholar
  12. 12.
    Orszag SA. Comparison of pseudospectral and spectral approximations.Stud Appl Math, 1972, 51: 253–259zbMATHGoogle Scholar
  13. 13.
    Fornberg B. A Practical Guide to Pseudospectral Methods. Cambridge: Cambridge University Press, 1996zbMATHGoogle Scholar
  14. 14.
    Walker JS. Fast Fourier Transforms. Florida: CRC Press, 1996zbMATHGoogle Scholar
  15. 15.
    Bellman R, Kashef BG, Casti J. Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations.J Comput Phys, 1972, 10: 40–52zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Wei GW. Discrete singular convolution for the solution of the Fokker-Planck equations.J Chem Phys, 1999, 110: 8930–8942CrossRefGoogle Scholar
  17. 17.
    Wei GW, Quasi wavelets and quasi interpolating wavelets.Chem Phys Lett, 1998, 296: 215–222CrossRefGoogle Scholar
  18. 18.
    Wei GW, Zhang DS, Kouri DJ, Hoffman DK. Lagrange distributed approximating functionals.Phys Rev Lett, 1997, 79: 775–779CrossRefGoogle Scholar
  19. 19.
    Schwartz L. Theore Des Distributions. Paris: Hermann, 1951Google Scholar
  20. 20.
    Korevaar J. Mathematical Methods. Volume I. New York: Academic Press, 1968zbMATHGoogle Scholar
  21. 21.
    Chui CK. An Introduction to Wavelets. San Diego: Academic Press, 1992zbMATHGoogle Scholar
  22. 22.
    Wei GW. Discrete singaular convolution method for the Sine-Gordon equation.Physica D, 2000, 137: 247–259zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Wei GW. Solving quantum eigenvalue problems by discrete singular convolution.J Phys B, 2000, 33: 343–352CrossRefGoogle Scholar
  24. 24.
    Wan DC, Wei GW. Numerical simulations of flows around multi-rectangular cylinders by DSCFSM. In: Proceedings of 4th International Conference on Hydrodynamics, September, 2000, Yokohama, JapanGoogle Scholar
  25. 25.
    Mead JL, Renaut RA. Optimal Runge-Kutta methods for first order pseudopsectral operators.J Comput Phys, 1999, 152: 404–419zbMATHMathSciNetCrossRefGoogle Scholar
  26. 26.
    Weinan E, Shu CW. A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow.J Comput Phys, 1994, 110: 39–46CrossRefGoogle Scholar
  27. 27.
    Bell J, Colella P, Glaz H. A second-order production method for the incompressible Navier-Stokes equations.J Comput Phys, 1989, 85: 257–283zbMATHMathSciNetCrossRefGoogle Scholar
  28. 28.
    Lou ZJ, Ferraro R. A parallel incompressible flow solver package with a parallel multigrid elliptic kernel.J Comput Phys, 1996, 125: 225–243zbMATHMathSciNetCrossRefGoogle Scholar
  29. 29.
    Ma YW, Fu DX, Kobayashi T, Taniguch N. Numerical solution of the incompressible Navier-Stokes equations with an upwind compact difference scheme.Int J Numer Meth Fluids, 1999, 30: 509–521zbMATHCrossRefGoogle Scholar

Copyright information

© Chinese Society of Theoretical and Applied Mechanics 2000

Authors and Affiliations

  • D. C. Wan
    • 1
  • G. W. Wei
    • 2
  1. 1.Shanghai Institute of Appl Math and MechShanghai UniversityShanghaiChina
  2. 2.Dept of Computational ScienceNational University of SingaporeSingapore

Personalised recommendations