Abstract
In this paper, a numerical technique for solving the regularised long wave equation (RLW) is presented using a wavelet Galerkin (WG) method in space and a fourth-order Runge–Kutta (RK) technique in time. We study the convergence analysis of the obtained numerical solutions and investigate the results for the motions of double and single solitary waves, undular bores and conservation properties of mass, energy and momentum in order to verify the applicability and performance of the proposed method. Simulation results are further compared with the known analytical solutions and some previous published numerical results. It is concluded that the present method remarkably improves the accuracy of the Galerkin-based methods for numerically solving a large class of nonlinear and weakly dispersive ocean waves.
Similar content being viewed by others
References
D H Peregrine, J. Fluid Mech. 25, 321 (1966)
T B Benjamin, J L Bona and J J Mahony, Philos. Trans. R. Soc. Lond. A 272(1220), 47 (1972)
Kh Abdulloev, H Bogalubsky and V G Markhankov, J. Phys. Lett. A 56, 427 (1976)
M Wadati, J. Phys. Lett. A 57(5–6), 841 (2001)
J C Eilbeck and G R McGuire, J. Comput. Phys. 19, 43 (1975)
J L Bona and A Soyeur, J. Nonlinear Sci. 4, 49 (1994)
K R Raslan, J. Appl. Math. Comput. 168, 795 (2005)
L R T Gardner, G A Gardner and A Dogan, Commun. Numer. Methods Eng. 12, 795 (1996)
D M Sloan, J. Comput. Appl. Math. 36(2), 159 (1991)
A Araujo and A Duran, J. Appl. Numer. Math. 36, 197 (2001)
A Duran and M A Lopez-Marcos, J. Phys. 36(28), 7761 (2003)
L R T Gardner and G A Gardner, J. Comput. Phys. 91, 441 (1990)
Z Luo and R Liu, SIAM J. Numer. Anal. 36, 89 (1998)
P C Jain, R Shankar and T V Singh, Commun. Numer. Methods Eng. 9, 579 (1993)
S I Zaki, J. Comput. Phys. Commun. 138, 80 (2001)
I Dag, B Saka and D Irk, J. Comput. Appl. Math. 190, 532 (2006)
Ö Oruç, F Bulut and A Esen, Mediterr. J. Math. 13(5), 3235 (2016)
Ö Oruç, F Bulut and A Esen, Pramana – J. Phys. 87(6): 94 (2016)
M Bakhoday-Paskyabi, Wave Motion 73, 24 (2017)
J Hozman and J Lamac, Bound. Value Probl. 2013, 1 (2013)
X Kang, K Cheng and C Guo, Adv. Diff. Equ. 339, 1 (2015)
I Daubechies, Commun. Pure Appl. Math. 41(7), 909 (1988)
I Daubechies, Ten lectures on wavelets (SIAM, Philadelphia, 1992)
T Kremp and W Freude, J. Ligthwave Technol. 23, 1491 (2005)
M Bakhoday-Paskyabi and F Rashidi, WSEAS Trans. Math. 4, 204 (2005)
D Antonopoulos, V Dougalis and D Mitsotakis, SIAM J. Numer. Anal. 55(2), 841 2017
V Thomee and B Wendroff, SIAM J. Numer. Anal. 11, 1039 (1974)
D Liang, Q Guo and S Gong, Commun. Comput. Phys. 6, 109 (2009)
A Doğan, J. Appl. Math. Model. 26, 771 (2002)
L R T Gardner, G A Gardner and I Dag, Commun. Numer. Methods Eng. 11, 59 (1995)
A Doğan, Commun. Numer. Methods Eng. 17, 485 (2001)
K R Raslan, J. Appl. Math. Comput. 167, 1101 (2005)
L R T Gardner, G A Gardner, F A Ayoub and N K Amein, Comput. Methods Appl. Mech. Eng. 147(1–2), 147 (1997)
B Saka and İ Dag, Numer. Methods Partial Diff. Eqs. 23(3), 731 (2007)
A Esen and S Kutluay, Appl. Math. Comput. 174(2), 833 (2006)
İ. Dag, A Doğan and B Saka, Int. J. Comput. Math. 80(6), 743 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bakhoday-Paskyabi, M., Valinejad, A. & Azodi, H.D. Numerical solution of regularised long ocean waves using periodised scaling functions. Pramana - J Phys 92, 71 (2019). https://doi.org/10.1007/s12043-019-1726-2
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12043-019-1726-2