Abstract
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.
Similar content being viewed by others
References
Peregrine, D. H. Calculations of the development of an undular bore. J. Fluid. Mech., 25(2), 321–330 (1996)
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)
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)
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)
Sanz-Serna, J. M. and Christie, I. Petrov-Galerkin methods for non-linear dispersive wave. J. Comput. Phys., 39, 94–102 (1981)
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)
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)
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)
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)
Guo, L. and Chen, H. H 1-Galerkin mixed finite element method for the regularized long wave equation. Computing, 77, 205–221 (2006)
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)
Cai, Z., Jones, J. E., Mccormick, S. F., and Russell, T. F. Control-volume mixed finite element methods. Comput. Geosci., 1, 289–315 (1997)
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)
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)
Yang, S. X. and Jiang, Z. W. Mixed covolume method for parabolic problems on triangular grids. Appl. Math. Comput., 215, 1251–1265 (2009)
Rui, H. X. Symmetric mixed covolume methods for parabolic problems. Numerical Methods for Partial Differential Equations, 18, 561–583 (2002)
Ciarlet, P. G. The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam (1978)
Author information
Authors and Affiliations
Corresponding author
Additional information
Project supported by the National Natural Science Fundation of China (No. 11061021), the Science Research of Inner Mongolia Advanced Education (Nos.NJ10006, NJ10016, and NJZZ12011), and the National Science Foundation of Inner Mongolia (Nos. 2011BS0102 and 2012MS0106)
Rights and permissions
About this article
Cite this article
Fang, Zc., Li, H. Numerical solutions to regularized long wave equation based on mixed covolume method. Appl. Math. Mech.-Engl. Ed. 34, 907–920 (2013). https://doi.org/10.1007/s10483-013-1716-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10483-013-1716-8