The Numerical Solution of the MRLW Equation Using the Multigrid Method
Affiliation(s)
1 Mathematics Department, Faculty of Education and Science (AL-Khurmah Branch), Taif University, Taif, KSA. 2 Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo, Egypt. 3 Mathematics Department, Faculty of Science, Suez Canal University, Ismailia, Egypt. 4 Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt.
1 Mathematics Department, Faculty of Education and Science (AL-Khurmah Branch), Taif University, Taif, KSA. 2 Mathematics Department, Faculty of Science, Al-Azhar University, Nasr City, Cairo, Egypt. 3 Mathematics Department, Faculty of Science, Suez Canal University, Ismailia, Egypt. 4 Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt.
ABSTRACT
In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation
, by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Using
error norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other techniques.
In this paper, we obtained the numerical solutions of the modified regularized long-wave (MRLW) equation
, by using the multigrid method and finite difference method. The solitary wave motion, interaction of two and three solitary waves, and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Usingerror norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other techniques.
Cite this paper
Essa, Y. , Abouefarag, I. , Rahmo, E. (2014) The Numerical Solution of the MRLW Equation Using the Multigrid Method. Applied Mathematics, 5, 3328-3334. doi: 10.4236/am.2014.521310.
Essa, Y. , Abouefarag, I. , Rahmo, E. (2014) The Numerical Solution of the MRLW Equation Using the Multigrid Method. Applied Mathematics, 5, 3328-3334. doi: 10.4236/am.2014.521310.
References
[1] Brandt, A. (1977) Multi-Level Adaptive Solution to Boundary-Value Problem. Mathematics of Computation, 31, 333-390.
http://dx.doi.org/10.1090/S0025-5718-1977-0431719-X [2] Goldstein, C.I. (1989) Analysis and Application of Multigrid Preconditioners for Singularly Perturbed Boundary Value Problems. SIAM Journal on Numerical Analysis, 26, 1090-1123.
http://dx.doi.org/10.1137/0726061 [3] Dennis, J.C. (1984) Multigrid Method for Partial Differential Equations. Studies in numerical Analysis, 24. [4] Hachbusch, W. (1984) Multigrid Methods and Applications. Springer, Berlin. [5] Brandt, A. (1988) Multilevel Computations. Marcel Dekker, New York. [6] Wesseling, P. (1992) An Introduction to Multigrid Methods. John Wiley & Sons, New York. [7] Hassen, N.A., Osama, E.M. and Fatema, H.M. (1995) The Relaxation Schemes for the Two Dimensional Anisotropic Partial Differential Equations Using Multigrid Method. 20th INI Conf. for Stat., Comp. Sci., Scient. & Social Appl., 3-9 April 1995, Cairo. [8] Abo Essa, Y.M., Amer, T.S. and Ibrahim, I.A. (2011) Numerical Treatment of the Generalized Regularized Long-Wave Equation. Far East Journal of Applied Mathematics, 52, 147-154. [9] Abo Essa, Y.M., Amer, T.S. and Abdul-Moniem, I.B. (2012) Application of Multigrid Technique for the Numerical Solution of the Non-Linear Dispersive Waves Equations. International Journal of Mathematical Archive, 3, 4903-4910. [10] Peregrine, D.H. (1966) Calculations of the Development of an Undular Bore. Journal of Fluid Mechanics, 25, 321-330.
http://dx.doi.org/10.1017/S0022112066001678 [11] Benjamin, T.B., Bona, J.L. and Mahony, J.L. (1972) Model Equations for Long Waves in Nonlinear Dispersive Systems. Philosophical Transactions of the Royal Society of London. Series A, 272, 47-78.
http://dx.doi.org/10.1098/rsta.1972.0032 [12] Gardner, L.R.T., Gardner, G.A., Ayub, F.A. and Amein, N.K. (1997) Approximations of Solitary Waves of the MRLW Equation by B-Spline Finite Element. Arabian Journal for Science and Engineering, 22, 183-193. [13] Khalifa, A.K., Raslan, K.R. and Alzubaidi, H.M. (2008) A Collocation Method with Cubic B-Splines for Solving the MRLW Equation. Journal of Computational and Applied Mathematics, 212, 406-418.
http://dx.doi.org/10.1016/j.cam.2006.12.029 [14] Khalifa, A.K., Raslan, K.R. and Alzubaidi, H.M. (2007) A Finite Difference Scheme for the MRLW and Solitary Wave Interactions. Applied Mathematics and Computation, 189, 346-354.
http://dx.doi.org/10.1016/j.amc.2006.11.104 [15] Raslan, K.R. (2009) Numerical Study of the Modified Regularized Long Wave (MRLW) Equation. Chaos, Solitons and Fractals, 42, 1845-1853.
http://dx.doi.org/10.1016/j.chaos.2009.03.098 [16] Raslan, K.R. and Hassan, S.M. (2009) Solitary Waves of the MRLW Equation. Applied Mathematics Letters, 22, 984-989.
http://dx.doi.org/10.1016/j.aml.2009.01.020 [17] Ali, A. (2009) Mesh Free Collocation Method for Numerical Solution of Initial-Boundary Value Problems Using Radial Basis Functions. Ph.D. Thesis, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Khyber Pakhtunkhwa. [18] Haq, F., Islam, S. and Tirmizi, I.A. (2010) A Numerical Technique for Solution of the MRLW Equation Using Quartic B-Splines. Applied Mathematical Modelling, 34, 4151-4160.
http://dx.doi.org/10.1016/j.apm.2010.04.012 [19] Battal Gazi Karakoc, S. and Geyikli, T. (2013) Petrov-Galerkin Finite Element Method for Solving the MRLW Equation. Mathematical Sciences, 7, 25.
[1] Brandt, A. (1977) Multi-Level Adaptive Solution to Boundary-Value Problem. Mathematics of Computation, 31, 333-390.
http://dx.doi.org/10.1090/S0025-5718-1977-0431719-X [2] Goldstein, C.I. (1989) Analysis and Application of Multigrid Preconditioners for Singularly Perturbed Boundary Value Problems. SIAM Journal on Numerical Analysis, 26, 1090-1123.
http://dx.doi.org/10.1137/0726061 [3] Dennis, J.C. (1984) Multigrid Method for Partial Differential Equations. Studies in numerical Analysis, 24. [4] Hachbusch, W. (1984) Multigrid Methods and Applications. Springer, Berlin. [5] Brandt, A. (1988) Multilevel Computations. Marcel Dekker, New York. [6] Wesseling, P. (1992) An Introduction to Multigrid Methods. John Wiley & Sons, New York. [7] Hassen, N.A., Osama, E.M. and Fatema, H.M. (1995) The Relaxation Schemes for the Two Dimensional Anisotropic Partial Differential Equations Using Multigrid Method. 20th INI Conf. for Stat., Comp. Sci., Scient. & Social Appl., 3-9 April 1995, Cairo. [8] Abo Essa, Y.M., Amer, T.S. and Ibrahim, I.A. (2011) Numerical Treatment of the Generalized Regularized Long-Wave Equation. Far East Journal of Applied Mathematics, 52, 147-154. [9] Abo Essa, Y.M., Amer, T.S. and Abdul-Moniem, I.B. (2012) Application of Multigrid Technique for the Numerical Solution of the Non-Linear Dispersive Waves Equations. International Journal of Mathematical Archive, 3, 4903-4910. [10] Peregrine, D.H. (1966) Calculations of the Development of an Undular Bore. Journal of Fluid Mechanics, 25, 321-330.
http://dx.doi.org/10.1017/S0022112066001678 [11] Benjamin, T.B., Bona, J.L. and Mahony, J.L. (1972) Model Equations for Long Waves in Nonlinear Dispersive Systems. Philosophical Transactions of the Royal Society of London. Series A, 272, 47-78.
http://dx.doi.org/10.1098/rsta.1972.0032 [12] Gardner, L.R.T., Gardner, G.A., Ayub, F.A. and Amein, N.K. (1997) Approximations of Solitary Waves of the MRLW Equation by B-Spline Finite Element. Arabian Journal for Science and Engineering, 22, 183-193. [13] Khalifa, A.K., Raslan, K.R. and Alzubaidi, H.M. (2008) A Collocation Method with Cubic B-Splines for Solving the MRLW Equation. Journal of Computational and Applied Mathematics, 212, 406-418.
http://dx.doi.org/10.1016/j.cam.2006.12.029 [14] Khalifa, A.K., Raslan, K.R. and Alzubaidi, H.M. (2007) A Finite Difference Scheme for the MRLW and Solitary Wave Interactions. Applied Mathematics and Computation, 189, 346-354.
http://dx.doi.org/10.1016/j.amc.2006.11.104 [15] Raslan, K.R. (2009) Numerical Study of the Modified Regularized Long Wave (MRLW) Equation. Chaos, Solitons and Fractals, 42, 1845-1853.
http://dx.doi.org/10.1016/j.chaos.2009.03.098 [16] Raslan, K.R. and Hassan, S.M. (2009) Solitary Waves of the MRLW Equation. Applied Mathematics Letters, 22, 984-989.
http://dx.doi.org/10.1016/j.aml.2009.01.020 [17] Ali, A. (2009) Mesh Free Collocation Method for Numerical Solution of Initial-Boundary Value Problems Using Radial Basis Functions. Ph.D. Thesis, Ghulam Ishaq Khan Institute of Engineering Sciences and Technology, Khyber Pakhtunkhwa. [18] Haq, F., Islam, S. and Tirmizi, I.A. (2010) A Numerical Technique for Solution of the MRLW Equation Using Quartic B-Splines. Applied Mathematical Modelling, 34, 4151-4160.
http://dx.doi.org/10.1016/j.apm.2010.04.012 [19] Battal Gazi Karakoc, S. and Geyikli, T. (2013) Petrov-Galerkin Finite Element Method for Solving the MRLW Equation. Mathematical Sciences, 7, 25.