High-order conservative difference scheme for a model of nonlinear dispersive equations
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High-order conservative difference scheme for a model of nonlinear dispersive equations Asma Rouatbi1 · Talha Achouri1 · Khaled Omrani1
Received: 1 February 2017 / Accepted: 23 December 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018
Abstract A high-order nonlinear conservative difference scheme method is proposed to solve a model of nonlinear dispersive equation: RLW-KdV equation. The existence of the solution was proved by the Brouwer fixed point theorem. The unconditional stability besides uniqueness of the difference scheme are also obtained. The convergence of the proposed method is proved to be fourth-order in space and second-order in time in the discrete L ∞ norm. An application on the RLW equation is discussed numerically in detail. Furthermore, interaction of solitary waves with different amplitudes are shown. The three invariants of the motion are evaluated to show the conservation properties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. At last some numerical examples are reported to confirm the theoretical results. Keywords Nonlinear RLW-KdV equation · Conservation · Existence · Uniqueness · Stability · Convergence Mathematics Subject Classification 65M06 · 65M12 · 65M15
1 Introduction The well-known regularized long-wave (RLW) equation u t − u x xt + u x + uu x = 0,
(1.1)
Communicated by Pierangelo Marcati.
B
Khaled Omrani [email protected] Asma Rouatbi [email protected] Talha Achouri [email protected]
1
Institut Supérieur des Sciences Appliquées et de Technologie de Sousse, 4003 Sousse Ibn Khaldoun, Tunisia
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A. Rouatbi et al.
and its counterpart, the Korteweg–de Vries (KdV) equation u t + u x x x + u x + uu x = 0,
(1.2)
were both proposed as model problems for long waves in nonlinear dispersive media. Equation (1.1) was originally introduced to describe the behavior of the undular bore (Peregrine 1966). It has also been derived from the study of water waves and ion acoustic plasma waves. An analytical solution for the RLW equation was found under the restricted initial and boundary conditions in Benjamin et al. (1972). The same for the KdV equation, it is found naturally il all kinds of models. Among them, there are internal gravity waves in a stratified fluid (highly relevant for geophysics), waves in an astrophysical plasma, electrical transmission line behavior and even blood pressure. Waves-soliton solutions of the KdV equation explain why our pulse, coming from a localized pressure wave in our arteries, is detectable all over our body and persists despite changes in local conditions and artery geometry in the circulation system. In the recent years, several numerical methods for the solution of the RLW equation have been developed, including the finite difference (FD) method (Kutluay and Esen 2006; Berikelashvili and Mirianashvili 2011; Rashid 2005; Talha et al. 2006), the Fourier pseudospectral (PS) method (Guo and Cao 1988), the B-spline finite element method (FEM), Sinc-collocation
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