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Spectral method for solving the equal width equation based on Chebyshev polynomials. (English) Zbl 1170.76339
Summary: A spectral solution of the equal width (EW) equation based on the collocation method using Chebyshev polynomials as a basis for the approximate solution has been studied. Test problems, including the migration of a single solitary wave with different amplitudes are used to validate this algorithm which is found to be accurate and efficient. The three invariants of the motion are evaluated to determine the conservation properties of the algorithm. The interaction of two solitary waves is seen to cause the creation of a source for solitary waves. Usually these are of small magnitude, but when the amplitudes of the two interacting waves are opposite, the source produces trains of solitary waves whose amplitudes are of the same order as those of the initial waves. The three invariants of the motion of the interaction of the three positive solitary waves are computed to determine the conservation properties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Comparisons are made with the most recent results both for the error norms and the invariant values.

MSC:
76M22 Spectral methods applied to problems in fluid mechanics
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
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[1] Peregrine, D.H.: Calculations of the development of an undular bore. J. Fluid Mech. 25, 321–330 (1966)
[2] Abdulloev, Kh.O., Bogolubsky, H., Makhankov, V.G.: One more example of inelastic soliton interaction. Phys. Lett. A 56, 427–428 (1976)
[3] Morrison, P.J., Meiss, J.D., Carey, J.R.: Scattering of RLW solitary waves. Physica D 11, 324–336 (1981) · Zbl 0599.76028
[4] 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
[5] Gardner, L.R.T., Gardner, G.A.: Solitary waves of the equal width wave equation. J. Comput. Phys. 101, 218–223 (1992) · Zbl 0759.65086
[6] Gardner, L.R.T., Gardner, G.A., Ayoub, F.A., Amein, N.K.: Simulations of the EWE undular bore. Commun. Num. Methods Eng. 13, 583–592 (1997) · Zbl 0883.76048
[7] Zaki, S.I.: A least-squares finite element scheme for the EW equation. Comput. Methods Appl. Mech. Eng. 189, 587–594 (2000) · Zbl 0963.76057
[8] Zaki, S.I.: Solitary waves induced by the boundary forced Ew equation. Comput. Methods Appl. Mech. Eng. 190, 4881–4887 (2001) · Zbl 1011.76048
[9] Raslan, K.R.: Collocation method using quartic B-spline for the equal width (EW) equation. Appl. Math. Comput. (USA) 168, 785–805 (2005) · Zbl 1082.65583
[10] Soliman, A.A.: Numerical simulation of the generalised regularised long wave equation by He’s variational iteration method. Math. Comput. Simul. 70(2), 119–124 (2005) · Zbl 1152.65467
[11] Soliman, A.A., Hussein, M.H.: Collocation solution for RLW equation with septic spline. Appl. Math. Comput. 161, 623–636 (2005) · Zbl 1061.65102
[12] Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge, UK (1995) · Zbl 0912.65091
[13] Olver, P.J.: Euler operators and conservation laws of the BBM equation. Math. Proc. Cambridge Phil. Soc. 85, 143–159 (1979) · Zbl 0387.35050
[14] Elibeck, J.C., McGuire, G.R.: Numerical study of the RLW equation II: Interaction of solitary waves. J. Comput. Phys. 23, 63–73 (1977) · Zbl 0361.65100
[15] Iskandar, L., El-Deen Mohamedein, M.Sh.: Solitary waves interaction for the BBM equation. Comput. Methods Appl. Mech. Eng. 96, 361–372 (2001).
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