## Abstract and Applied Analysis

### Explicit Multistep Mixed Finite Element Method for RLW Equation

#### Abstract

An explicit multistep mixed finite element method is proposed and discussed for regularized long wave (RLW) equation. The spatial direction is approximated by the mixed Galerkin method using mixed linear space finite elements, and the time direction is discretized by the explicit multistep method. The optimal error estimates in ${L}^{2}$ and ${H}^{1}$ norms for the scalar unknown $u$ and its flux $q={u}_{x}$ based on time explicit multistep method are derived. Some numerical results are given to verify our theoretical analysis and illustrate the efficiency of our method.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 768976, 12 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393511906

Digital Object Identifier
doi:10.1155/2013/768976

Mathematical Reviews number (MathSciNet)
MR3064335

Zentralblatt MATH identifier
1275.65061

#### Citation

Liu, Yang; Li, Hong; Du, Yanwei; Wang, Jinfeng. Explicit Multistep Mixed Finite Element Method for RLW Equation. Abstr. Appl. Anal. 2013 (2013), Article ID 768976, 12 pages. doi:10.1155/2013/768976. https://projecteuclid.org/euclid.aaa/1393511906

#### References

• D. H. Peregrine, “Calculations of the development of an undular bore,” Journal of Fluid Mechanics, vol. 25, no. 2, pp. 321–330, 1966.
• T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Philosophical Transactions of the Royal Society of London A, vol. 272, no. 1220, pp. 47–78, 1972.
• P. J. Olver, “Euler operators and conservation laws of the BBM equation,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 85, no. 1, pp. 143–160, 1979.
• B. Saka and I. Dağ, “Quartic B-spline collocation algorithms for numerical solution of the RLW equation,” Numerical Methods for Partial Differential Equations, vol. 23, no. 3, pp. 731–751, 2007.
• J. L. Bona and P. J. Bryant, “A mathematical model for long waves generated by wavemakers in non-linear dispersive systems,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 73, no. 5, pp. 391–405, 1973.
• D. Bhardwaj and R. Shankar, “A computational method for regularized long wave equation,” Computers & Mathematics with Applications, vol. 40, no. 12, pp. 1397–1404, 2000.
• X. H. Zhao, D. S. Li, and D. M. Shi, “A finite difference scheme for RLW-Burgers equation,” Journal of Applied Mathematics & Informatics, vol. 26, no. 3-4, pp. 573–581, 2008.
• L. Zhang, “A finite difference scheme for generalized regularized long-wave equation,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 962–972, 2005.
• I. Dağ, A. Dogan, and B. Saka, “B-spline collocation methods for numerical solutions of the RLW equation,” International Journal of Computer Mathematics, vol. 80, no. 6, pp. 743–757, 2003.
• A. Esen and S. Kutluay, “Application of a lumped Galerkin method to the regularized long wave equation,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 833–845, 2006.
• I. Dağ, B. Saka, and D. Irk, “Galerkin method for the numerical solution of the RLW equation using quintic B-splines,” Journal of Computational and Applied Mathematics, vol. 190, no. 1-2, pp. 532–547, 2006.
• K. R. Raslan, “A computational method for the regularized long wave (RLW) equation,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1101–1118, 2005.
• L. Wahlbin, “Error estimates for a Galerkin method for a class of model equations for long waves,” Numerische Mathematik, vol. 23, pp. 289–303, 1975.
• Z. D. Luo and R. X. Liu, “Mixed finite element analysis and numerical solitary solution for the RLW equation,” SIAM Journal on Numerical Analysis, vol. 36, no. 1, pp. 89–104, 1999.
• L. Guo and H. Chen, “${H}^{1}$-Galerkin mixed finite element method for the regularized long wave equation,” Computing, vol. 77, no. 2, pp. 205–221, 2006.
• H. M. Gu and N. Chen, “Least-squares mixed finite element methods for the RLW equations,” Numerical Methods for Partial Differential Equations, vol. 24, no. 3, pp. 749–758, 2008.
• J. L. Bona, W. G. Pritchard, and L. R. Scott, “Numerical schemes for a model for nonlinear dispersive waves,” Journal of Computational Physics, vol. 60, no. 2, pp. 167–186, 1985.
• S. U. Islam, S. Haq, and A. Ali, “A meshfree method for the numerical solution of the RLW equation,” Journal of Computational and Applied Mathematics, vol. 223, no. 2, pp. 997–1012, 2009.
• D. Kaya, “A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 833–841, 2004.
• L. Q. Mei and Y. P. Chen, “Explicit multistep method for the numerical solution of RLW equation,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9547–9554, 2012.
• L. R. T. Gardner, G. A. Gardner, and A. Dogan, “A least-squares finite element scheme for the RLW equation,” Communications in Numerical Methods in Engineering, vol. 12, no. 11, pp. 795–804, 1996.
• I. Dağ, “Least squares quadratic B-spline finite element method for the regularized long wave equation,” Computer Methods in Applied Mechanics and Engineering, vol. 182, no. 1-2, pp. 205–215, 2000.
• I. Dağ and M. N. Özer, “Approximation of the RLW equation by the least square cubic B-spline finite element method,” Applied Mathematical Modelling, vol. 25, no. 3, pp. 221–231, 2001.
• A. Dogan, “Numerical solution of RLW equation using linear finite elements within Galerkin's method,” Applied Mathematical Modelling, vol. 26, no. 7, pp. 771–783, 2002.
• P. Chatzipantelidis, “Explicit multistep methods for nonstiff partial differential equations,” Applied Numerical Mathematics, vol. 27, no. 1, pp. 13–31, 1998.
• G. Akrivis, O. Karakashian, and F. Karakatsani, “Linearly implicit methods for nonlinear evolution equations,” Numerische Mathematik, vol. 94, no. 3, pp. 403–418, 2003.
• A. K. Pani, R. K. Sinha, and A. K. Otta, “An ${H}^{1}$-Galerkin mixed method for second order hyperbolic equations,” International Journal of Numerical Analysis and Modeling, vol. 1, no. 2, pp. 111–129, 2004.
• M. F. Wheeler, “A priori ${L}_{2}$ error estimates for Galerkin approximations to parabolic partial differential equations,” SIAM Journal on Numerical Analysis, vol. 10, pp. 723–759, 1973.
• J. L. Bona, W. G. Pritchard, and L. R. Scott, “An evaluation of a model equation for water waves,” Philosophical Transactions of the Royal Society of London A, vol. 302, no. 1471, pp. 457–510, 1981.