Applied Mathematics and Mechanics

, Volume 34, Issue 7, pp 907–920 | Cite as

Numerical solutions to regularized long wave equation based on mixed covolume method

  • Zhi-chao Fang (方志朝)
  • Hong Li (李 宏)Email author


The mixed covolume method for the regularized long wave equation is developed and studied. By introducing a transfer operator γ h , which maps the trial function space into the test function space, and combining the mixed finite element with the finite volume method, the nonlinear and linear Euler fully discrete mixed covolume schemes are constructed, and the existence and uniqueness of the solutions are proved. The optimal error estimates for these schemes are obtained. Finally, a numerical example is provided to examine the efficiency of the proposed schemes.

Key words

regularized long wave equation mixed covolume method fully discrete optimal error estimate 

Chinese Library Classification


2010 Mathematics Subject Classification

65M08 65M12 65M60 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Peregrine, D. H. Calculations of the development of an undular bore. J. Fluid. Mech., 25(2), 321–330 (1996)CrossRefGoogle Scholar
  2. [2]
    Benjamin, T. B., Bona, J. L., and Mahony, J. J. Model equations for long waves in non-linear dispersive systems. Philos. Trans. R. Soc. London Ser. A, 272, 47–48 (1972)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Elibeck, J. C. and McGuire, G. R. Numerical study of the RLW equation II: interaction of solitary waves. J. Comput. Phys., 23, 63–73 (1977)CrossRefGoogle Scholar
  4. [4]
    Alexander, M. E. and Morris, J. L. Galerkin methods applied to some model equations for nonlinear dispersive waves. J. Comput. Phys., 30, 428–451 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    Sanz-Serna, J. M. and Christie, I. Petrov-Galerkin methods for non-linear dispersive wave. J. Comput. Phys., 39, 94–102 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Guo, B. Y. and Cao, W. M. The Fourier pseudospectral method with a restrain operator for the RLW equation. J. Comput. Phys., 74, 110–126 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Gardner, L. R. T., Gardner, G. A., and Dag, I. A B-spline finite element method for the regularized long wave equation. Commum. Numer. Meth. Eng., 11, 59–68 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Dag, I., Saka, B., and Irk, D. Galerkin method for the numerical solution of the RLW equation using quintic B-splines. J. Comput. Appl. Math., 190, 532–547 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Luo, Z. D. and Liu, R. X. Mixed finite element analysis and numerical solitary solution for the RLW equation. SIAM J. Numer. Anal., 36, 189–204 (1999)MathSciNetGoogle Scholar
  10. [10]
    Guo, L. and Chen, H. H 1-Galerkin mixed finite element method for the regularized long wave equation. Computing, 77, 205–221 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    Gu, H. M. and Chen, N. Least-squares mixed finite element methods for the RLW equations. Numerical Methods for Partial Differential Equations, 24(3), 749–758 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    Cai, Z., Jones, J. E., Mccormick, S. F., and Russell, T. F. Control-volume mixed finite element methods. Comput. Geosci., 1, 289–315 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Jones, J. E. A Mixed Finite Volume Element Method for Accurate Computation of Fluid Velocities in Porous Media, Ph. D. dissertation, University of Colorado, Denver (1995)Google Scholar
  14. [14]
    Chou, S. H., Kwak, D. Y., and Vassilevski, P. S. Mixed covolume methods for the elliptic problems on triangular grids. SIAM J. Numer. Anal., 35, 1850–1861 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Yang, S. X. and Jiang, Z. W. Mixed covolume method for parabolic problems on triangular grids. Appl. Math. Comput., 215, 1251–1265 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    Rui, H. X. Symmetric mixed covolume methods for parabolic problems. Numerical Methods for Partial Differential Equations, 18, 561–583 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    Ciarlet, P. G. The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam (1978)zbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.School of Mathematical SciencesInner Mongolia UniversityHohhotP. R. China

Personalised recommendations