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Explicit finite difference methods for the EW and RLW equations. (English) Zbl 1102.65092
Summary: An extensive assessment of the accuracy of explicit finite difference methods for the solution of the equal-width (EW) and regularized long-wave (RLW) equations is reported. Such an assessment is based on the three invariants of these equations as well as on the magnitude of the errors of the numerical solution and has been performed as a function of the time step and grid spacing.
Two of the methods presented here make use of three-point, fourth-order accurate, finite difference formulae for the first- and second-order spatial derivatives. Two methods are based on the analytical solution of second-order ordinary differential equations which have locally exponential solutions, and the fourth technique is a standard finite difference scheme.
A linear stability analysis of the four methods is presented. It is shown that, for the EW and RLW equations, a compact operator method is more accurate than locally exponential techniques that make use of compact operator approximations. The latter are reported to be more accurate than exponential techniques that employ second-order accurate approximations, and, these, in turn, are more accurate than the standard explicit method.

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L75 Higher-order nonlinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Benjamin, T.B.; Bona, J.L.; Mahony, J.J., Model equations for long waves in nonlinear dispersive systems, Phil. trans. roy. soc. (London), series A, 272, 47-78, (1972) · Zbl 0229.35013
[2] Peregrine, D.H., Calculations of the development of an undular bore, J. fluid mech., 25, 321-330, (1966)
[3] Peregrine, D.H., Long waves on a beach, J. fluid mech., 27, 815-827, (1967) · Zbl 0163.21105
[4] Dodd, R.K.; Eilbeck, J.C.; Gibbon, J.D.; Morris, H.C., Solitons and nonlinear wave equations, (1982), Academic Press New York · Zbl 0496.35001
[5] Lewis, J.C.; Tjon, J.A., Resonant production of solitons in the RLW equation, Phys. lett. A, 73, 275-279, (1979)
[6] Raslan, K.R., Collocation method using quartic B-spline for the equal width (EW) equation, Appl. math. comput., 168, 795-805, (2005) · Zbl 1082.65583
[7] 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
[8] Gardner, L.R.T.; Gardner, G.A., Solitary waves of the regularised long-wave equation, J. comput. phys., 91, 441-459, (1990) · Zbl 0717.65072
[9] Sloan, D.M., Fourier pseudospectral solution of the regularised long wave equation, J. comput. appl. math., 36, 159-179, (1991) · Zbl 0732.65096
[10] Araújo, A.; Durán, A., Error propagation in the numerical integration of solitary waves. the regularized long wave equation, Appl. numer. math., 36, 197-217, (2001) · Zbl 0972.65062
[11] Durán, A.; López-Marcos, M.A., Conservative numerical methods for solitary wave interactions, J. phys. A, 36, 7761-7770, (2003) · Zbl 1038.35091
[12] Bona, J.L.; Soyeur, A., On the stability of solitary wave solutions of model equations for long waves, J. nonlinear sci., 4, 449-470, (1994) · Zbl 0809.35095
[13] Luo, Z.; Liu, R., Mixed finite element analysis and numerical solution for the RLW equation, SIAM J. numer. anal., 36, 89-104, (1998) · Zbl 0927.65123
[14] Jain, P.C.; Shankar, R.; Singh, T.V., Numerical solution of the regularized long-wave equation, Commun. numer. meth. eng., 9, 579-586, (1993) · Zbl 0779.65062
[15] Zaki, S.I., Solitary waves of the splitted RLW equation, Comput. phys. commun., 138, 80-91, (2001) · Zbl 0984.65103
[16] I. Dağ, B. Saka, D. Irk, Galerkin method for the numerical solution of the RLW equation using quintic B-splines, J. Comput. Appl. Math., in press. · Zbl 1086.65094
[17] El-Danaf, T.S.; Ramadan, M.A.; Abd Alaal, F.E.I., The use of Adomian decomposition method for solving the regularized long-wave equation, Chaos solitons fract., 26, 747-757, (2005) · Zbl 1073.35010
[18] Kaya, D.; El-Sayed, S.M., An application of the decomposition method for the generalized KdV and RLW equations, Chaos solitons fract., 17, 869-877, (2003) · Zbl 1030.35139
[19] Shivamoggi, B.K.; Rollins, D.K., Evolution of solitary-wave solution of the perturbed regularized long-wave equation, Chaos solitons fract., 13, 1129-1136, (2002) · Zbl 1031.76012
[20] Ramos, J.I., Implicit, compact, linearized θ-methods with factorization for multidimensional reaction-diffusion equations, Appl. math. comput., 94, 17-43, (1998) · Zbl 0943.65098
[21] Ramos, J.I., On diffusive methods and exponentially-fitted techniques, Appl. math. comput., 103, 69-96, (1999) · Zbl 0929.65057
[22] Ramos, J.I., A smooth locally-analytical technique for singularly perturbed two-point boundary value problems, Appl. math. comput., 163, 1123-1142, (2005) · Zbl 1067.65074
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