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Application of a lumped Galerkin method to the regularized long wave equation. (English) Zbl 1090.65114
Summary: A lumped Galerkin method based on quadratic B-spline finite elements is used to find numerical solutions of the one-dimensional regularized long wave equation with a variant of initial and boundary conditions. The obtained numerical results show that the present method is a remarkably successful numerical technique for solving the equation. Results are compared with published numerical solutions. A linear stability analysis of the scheme is also investigated.

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L75 Higher-order nonlinear hyperbolic equations
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[1] Peregrine, D.H., Calculations of the development of an undular bore, J. fluid mech., 25, 321-330, (1966)
[2] Benjamin, T.B.; Bona, J.L.; Mahony, J.J., Model equations for long waves in non-linear dispersive systems, Philos. trans. R. soc., London A, 272, 47-78, (1972) · Zbl 0229.35013
[3] Abdulloev, Kh.O.; Bogolubsky, H.; Markhankov, V.G., One more example of inelastic soliton interaction, Phys. lett. A, 56, 427-428, (1976)
[4] Eilbeck, J.C.; McGuire, G.R., Numerical study of the regularized long wave equation II: interaction of solitary wave, J. comp. phys., 23, 63-73, (1977) · Zbl 0361.65100
[5] Jain, P.C.; Shankar, R.; Singh, T.V., Numerical solution of regularized long-wave equation, Commun. numer. meth. eng., 9, 579-586, (1993) · Zbl 0779.65062
[6] Bhardwaj, D.; Shankar, R., A computational method for regularized long wave equation, Comp. math. appl., 40, 1397-1404, (2000) · Zbl 0965.65108
[7] 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., 93, 360-375, (1995) · Zbl 0739.76037
[8] Alexander, M.E.; Morris, J.L., Galerkin method applied to some model equations for nonlinear dispersive waves, J. comput. phys., 30, 428-451, (1979) · Zbl 0407.76014
[9] Sanz Serna, J.M.; Christie, I., Petrov Galerkin methods for non linear dispersive wave, J. comput. phys., 39, 94-102, (1981) · Zbl 0451.65086
[10] Gardner, L.R.T.; Gardner, G.A., Solitary waves of the regularized long-wave equation, J. comput. phys., 91, 441-459, (1990) · Zbl 0717.65072
[11] Gardner, L.R.T.; Gardner, G.A.; Dag, I., A B-spline finite element method for the regularized long wave equation, Commun. numer. meth. eng., 11, 59-68, (1995) · Zbl 0819.65125
[12] Gardner, L.R.T.; Gardner, G.A.; Dogan, A., A least-squares finite element scheme for the RLW equation, Commun. numer. meth. eng., 12, 795-804, (1996) · Zbl 0867.76040
[13] Dag, I.; Özer, M.N., Approximation of RLW equation by least square cubic B-spline finite element method, Appl. math. model., 25, 221-231, (2001) · Zbl 0990.65110
[14] Dogan, A., Numerical solution of RLW equation using linear finite elements within galerkin’s method, Appl. math. model., 26, 771-783, (2002) · Zbl 1016.76046
[15] Zaki, S.I., Solitary waves of the splitted RLW equation, Comput. phys. commun., 138, 80-91, (2001) · Zbl 0984.65103
[16] Dag, I.; Saka, B.; Irk, D., Application of cubic B-splines for numerical solution of the RLW equation, Appl. math. comput., 159, 373-389, (2004) · Zbl 1060.65110
[17] Soliman, A.A.; Raslan, K.R., Collocation method using quadratic B-spline for the RLW equation, Int. J. comput math., 78, 399-412, (2001) · Zbl 0990.65116
[18] Prenter, P.M., Splines and variational methods, (1975), Wiley New York · Zbl 0344.65044
[19] Smith, G.D., Numerical solution of partial differential equations: finite differenc methods, (1987), Clarendon Press Oxford
[20] Olver, P.J., Euler operators and conservation laws of the BBM equation, Math. proc. camb. phil. soc., 85, 143-160, (1979) · Zbl 0387.35050
[21] Bona, J.L.; Pritchard, W.G.; Scott, L.R., Numerical schemes for a model of nonlinear dispersive waves, J. comp. phys., 60, 167-196, (1985) · Zbl 0578.65120
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