Abstract
We propose four conservative schemes for the regularized long-wave (RLW) equation. The RLW equation has three invariants: mass, momentum, and energy. Our schemes are designed by using the discrete variational derivative method to inherit appropriate conservation properties from the equation. Two of our schemes conserve mass and momentum, while the other two schemes conserve mass and energy. With one of our schemes, we prove the numerical solution stability, the existence of the solutions, and the convergence of the solutions. Through some numerical computation examples, we demonstrate the efficiency and robustness of our schemes.
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T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. London Ser. A,272 (1972), 47–78.
I. Dag and M.N. Ozer, Approximation of the RLW equation by the least square cubic B-spline finite element method. Appl. Math. Model.,25 (2001), 221–231.
I. Dag, B. Saka and D. Irk, Application of cubic B-splines for numerical solution of the RLW equation. Appl. Math. Comput.,159 (2004), 373–389.
A. Durán and M.A. López-Marcos, Conservative numerical methods for solitary wave inter-actions. J. Phys. A: Math. Gen.,36 (2003), 7761–7770.
J.C. Eilbeck and G.R. McGuire, Numerical study of the regularized long-wave equation, I: numerical methods. J. Comput. Phys.,19 (1975), 43–57.
J.C. Eilbeck and G.R. McGuire, Numerical study of the regularized long-wave equation, II: interaction of solitary waves. J. Comput. Phys.,23 (1977), 63–73.
D. Furihata, Finite difference schemes forδu/δt = (δ/x/gd)α δu/δG that inherit energy conservation or dissipation property. J. Comput. Phys.,156 (1999), 181–205.
D. Furihata, Finite difference schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math.,134 (2001), 37–57.
D. Furihata and T. Matsuo, A stable, convergent, conservative and linear finite difference scheme for the Cahn-Hilliard equation. Japan J. Indust. Appl. Math.,20 (2003), 65–85.
D. Furihata and M. Mori, A stable finite difference scheme for the Cahn-Hilliard equation based on the Lyapunov functional. ZAMM Z. angew. Math. Mech.,76 (1996), 405–406.
T. Hanada, N. Ishimura and M. Nakamura, Stable finite difference scheme for a model equation of phase separation. Appl. Math. Comput.,151 (2004), 95–104.
T. Ide, C. Hirota and M. Okada, Generalized energy integral forδu/δt = δG/δu and its finite difference schemes by means of discrete variational method and an application to Fujita problem. Adv. Math. Sci. Appl.,12 (2002), 755–778.
F. John, Lectures on Advanced Numerical Analysis. Gordon and Breach, New York, 1967.
T. Matsuo and D. Furihata, Dissipative or conservative finite difference schemes for complex-valued nonlinear partial differential equations. J. Comput. Phys.,171 (2001), 425–447.
T. Matsuo, M. Sugihara and M. Mori, A derivation of a finite difference scheme for the nonlinear Schrödinger equation by the discrete variational derivative (in Japanese). Trans. Japan Soc. Indust. Appl. Math.,8 (1998), 405–426.
P.J. Olver, Euler operators and conservation laws of the BBM equation. Math. Proc. Camb. Phil. Soc.,85 (1979), 143–160.
D.H. Peregrine, Calculations of the development of an undular bore. J. Fluid Mech.,25 (1966), 321–330.
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Koide, S., Furihata, D. Nonlinear and linear conservative finite difference schemes for regularized long wave equation. Japan J. Indust. Appl. Math. 26, 15 (2009). https://doi.org/10.1007/BF03167544
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DOI: https://doi.org/10.1007/BF03167544