Petrov-Galerkin finite element method for solving the MRLW equation. (English) Zbl 1295.65113
Summary: In this article, a Petrov-Galerkin method, in which the element shape functions are cubic and weight functions are quadratic B-splines, is introduced to solve the modified regularized long wave (MRLW) equation. 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. Accuracy and efficiency of the method are demonstrated by computing the numerical conserved laws and \(L_{2}\), \(L_\infty\) error norms. The computed results show that the present scheme is a successful numerical technique for solving the MRLW equation. A linear stability analysis based on the Fourier method is also investigated.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65D07 | Numerical computation using splines |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74J35 | Solitary waves in solid mechanics |
| 76B25 | Solitary waves for incompressible inviscid fluids |
References:
| [1] | Peregrine DH: Calculations of the development of an undular bore.J. Fluid Mech 1966, 25: 321-330. 10.1017/S0022112066001678 · doi:10.1017/S0022112066001678 |
| [2] | Benjamin TB, Bona JL, Mahoney JL: Model equations for long waves in nonlinear dispersive media.Phil. Trans. Roy. Soc. Lond. A 1972, 272: 47-78. 10.1098/rsta.1972.0032 · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032 |
| [3] | Eilbeck JC, McGuire GR: Numerical study of the regularized long wave equation, II: interaction of solitary wave.J. Comput. Phys 1977, 23: 63-73. 10.1016/0021-9991(77)90088-2 · Zbl 0361.65100 · doi:10.1016/0021-9991(77)90088-2 |
| [4] | Jain PC, Shankar R, Singh TV: Numerical solution of regularized long wave equation.Commun. Numer. Meth. Eng 1993, 9: 579-586. 10.1002/cnm.1640090705 · Zbl 0779.65062 · doi:10.1002/cnm.1640090705 |
| [5] | Bhardwaj D, Shankar R: A computational method for regularized long wave equation.Comput. Math. Appl 2000, 40: 1397-1404. · Zbl 0965.65108 · doi:10.1016/S0898-1221(00)00248-0 |
| [6] | Chang Q, Wang G, Guo B: Conservative scheme for a model of nonlinear dispersive waves and its solitary waves induced by boundary motion.J. Comput. Phys 1995, 93: 360-375. · Zbl 0739.76037 · doi:10.1016/0021-9991(91)90189-R |
| [7] | Gardner LRT, Gardner GA: Solitary waves of the regularized long wave equation.J. Comput. Phys 1990, 91: 441-459. 10.1016/0021-9991(90)90047-5 · Zbl 0717.65072 · doi:10.1016/0021-9991(90)90047-5 |
| [8] | Gardner LRT, Gardner GA, Dogan A: A least-squares finite element scheme for the RLW equation.Commun. Numer. Meth. Eng 1996, 12: 795-804. 10.1002/(SICI)1099-0887(199611)12:11<795::AID-CNM22>3.0.CO;2-O · Zbl 0867.76040 · doi:10.1002/(SICI)1099-0887(199611)12:11<795::AID-CNM22>3.0.CO;2-O |
| [9] | Gardner LRT, Gardner GA, Dag I: A B-spline finite element method for the regularized long wave equation.Commun. Numer. Meth. Eng 1995, 11: 59-68. 10.1002/cnm.1640110109 · Zbl 0819.65125 · doi:10.1002/cnm.1640110109 |
| [10] | Alexander ME, Morris JL: Galerkin method applied to some model equations for nonlinear dispersive waves.J. Comput. Phys 1979, 30: 428-451. 10.1016/0021-9991(79)90124-4 · Zbl 0407.76014 · doi:10.1016/0021-9991(79)90124-4 |
| [11] | Sanz Serna JM, Christie I: Petrov Galerkin methods for nonlinear dispersive wave.J. Comput. Phys 1981, 39: 94-102. 10.1016/0021-9991(81)90138-8 · Zbl 0451.65086 · doi:10.1016/0021-9991(81)90138-8 |
| [12] | Dogan A: Numerical solution of RLW equation using linear finite elements within Galerkin’s method.Appl. Math. Model 2002, 26: 771-783. 10.1016/S0307-904X(01)00084-1 · Zbl 1016.76046 · doi:10.1016/S0307-904X(01)00084-1 |
| [13] | Esen A, Kutluay S: Application of lumped Galerkin method to the regularized long wave equation.Appl. Math. Comput 2005, 174: 883-845. · Zbl 1090.65114 |
| [14] | Soliman AA, Raslan KR: Collocation method using quadratic B-spline for the RLW equation.Int. J. Comput. Math 2001, 78: 399-412. 10.1080/00207160108805119 · Zbl 0990.65116 · doi:10.1080/00207160108805119 |
| [15] | Soliman AA, Hussien MH: Collocation solution for RLW equation with septic spline.Appl. Math. Comput 2005, 161: 623-636. 10.1016/j.amc.2003.12.053 · Zbl 1061.65102 · doi:10.1016/j.amc.2003.12.053 |
| [16] | Raslan KR: A computational method for the regularized long wave (RLW) equation.Appl. Math. Comput 2005, 167: 1101-1118. 10.1016/j.amc.2004.06.130 · Zbl 1082.65582 · doi:10.1016/j.amc.2004.06.130 |
| [17] | Saka B, Dag I, Dogan A: Galerkin method for the numerical solution of the RLW equation using quadratic B-splines.Int. J. Comput. Math 2004,81(6):727-739. 10.1080/00207160310001650043 · Zbl 1060.65109 · doi:10.1080/00207160310001650043 |
| [18] | Dag I, Saka B, Irk D: Application of cubic B-splines for numerical solution of the RLW equation.Appl. Math. Comput 2004, 159: 373-389. 10.1016/j.amc.2003.10.020 · Zbl 1060.65110 · doi:10.1016/j.amc.2003.10.020 |
| [19] | Dag I, Ozer MN: Approximation of RLW equation by least-square cubic B-spline finite element method.Appl. Math. Model 2001, 25: 221-231. 10.1016/S0307-904X(00)00030-5 · Zbl 0990.65110 · doi:10.1016/S0307-904X(00)00030-5 |
| [20] | Zaki SI: Solitary waves of the splitted RLW equation.Comput. Phys. Commun 2001, 138: 80-91. 10.1016/S0010-4655(01)00200-4 · Zbl 0984.65103 · doi:10.1016/S0010-4655(01)00200-4 |
| [21] | Gou BY, Cao WM: The Fourier pseudo-spectral method with a restrain operator for the RLW equation.J. Comput. Phys 1988, 74: 110-126. 10.1016/0021-9991(88)90072-1 · Zbl 0684.65097 · doi:10.1016/0021-9991(88)90072-1 |
| [22] | Battal GaziKarakoc S: Numerical solutions of the modified equal width wave equation with finite elements method. Malatya: PhD Thesis, Inonu University; 2011. |
| [23] | Battal Gazi Karakoc S, Geyikli T: Numerical solution of the modified equal width wave equation.Int. J Differential Equations2012(2012): 10.155/2012/587208 · Zbl 1237.65110 |
| [24] | Geyikli T, Battal Gazi Karakoc S: Septic B-spline collocation method for the numerical solution of the modified equal width wave equation.Appl. Math 2011,6(2):739-749. · doi:10.4236/am.2011.26098 |
| [25] | Geyikli T, Battal Gazi Karakoc S: Petrov-galerkin method with cubic B-splines for solving the MEW equation.Bull. Belg. Math. Soc 2012, 19: 225-227. · Zbl 1245.65126 |
| [26] | Gardner LRT, Gardner GA, Geyikli T: The boundary forced MKdV equation.J. comput. Phys 1994, 11: 5-12. · Zbl 0806.65121 · doi:10.1006/jcph.1994.1113 |
| [27] | Ramos JI: Solitary wave interactions of the GRLW equation.Chaos, Solitons Fractals 2007, 33: 479-491. 10.1016/j.chaos.2006.01.016 · doi:10.1016/j.chaos.2006.01.016 |
| [28] | Zhang L: A finite difference scheme for generalized long wave equation.Appl. Math. Comput 2005,168(2):962-972. 10.1016/j.amc.2004.09.027 · Zbl 1080.65079 · doi:10.1016/j.amc.2004.09.027 |
| [29] | Kaya D, El-Sayed SM: An application of the decomposition method for the generalized KdV and RLW equations.Chaos, Solitons Fractals 2003, 17: 869-877. 10.1016/S0960-0779(02)00569-6 · Zbl 1030.35139 · doi:10.1016/S0960-0779(02)00569-6 |
| [30] | Roshan T: A Petrov-Galerkin method for solving the generalized regularized long wave (GRLW) equation.Comput. Math. Appl 2012, 63: 943-956. · Zbl 1247.65129 · doi:10.1016/j.camwa.2011.11.059 |
| [31] | Gardner LRT, Gardner GA, Ayoup FA, Amein NK: Simulations of solitary waves of the MRLW equation by B-spline finite element.Arab. J. Sci. Eng 1997, 22: 183-193. · Zbl 0893.35113 |
| [32] | Khalifa AK, Raslan KR, Alzubaidi HM: A collocation method with cubic B- splines for solving the MRLW equation.J. Comput. Appl. Math 2008, 212: 406-418. 10.1016/j.cam.2006.12.029 · Zbl 1133.65085 · doi:10.1016/j.cam.2006.12.029 |
| [33] | Khalifa AK, Raslan KR, Alzubaidi HM: A finite difference scheme for the MRLW and solitary wave interactions.Appl. Math. Comput 2007, 189: 346-354. 10.1016/j.amc.2006.11.104 · Zbl 1123.65085 · doi:10.1016/j.amc.2006.11.104 |
| [34] | Raslan KR: Numerical study of the modified regularized long wave equation.Chaos, Solitons Fractals 2009, 42: 1845-1853. 10.1016/j.chaos.2009.03.098 · Zbl 1198.65170 · doi:10.1016/j.chaos.2009.03.098 |
| [35] | Raslan KR, Hassan SM: Solitary waves for the MRLW equation.Appl. Math. Lett 2009, 22: 984-989. 10.1016/j.aml.2009.01.020 · Zbl 1179.65126 · doi:10.1016/j.aml.2009.01.020 |
| [36] | Haq F, Islam S, Tirmizi IA: A numerical technique for solution of the MRLW equation using quartic B-splines.Appl. Math. Model 2fig1, 34: 4151-4160. · Zbl 1201.65182 · doi:10.1016/j.apm.2010.04.012 |
| [37] | Ali, A., Mesh free collocation method for numerical solution of initial-boundary value problems using radial basis functions (2009) |
| [38] | Khalifa AK, Raslan KR, Alzubaidi HM: Numerical study using ADM for the modified regularized long wave equation.Appl. Math. Model 2008, 32: 2962-2972. 10.1016/j.apm.2007.10.014 · Zbl 1167.65450 · doi:10.1016/j.apm.2007.10.014 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
