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Two conservative difference schemes for a model of nonlinear dispersive equations

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  • Rouatbi, Asma
  • Omrani, Khaled

Abstract

Two conservative differences schemes for the nonlinear dispersive Benjamin–Bona–Mahony–KdV (BBM-KdV) equation are proposed. The first scheme is two-level and nonlinear-implicit. The second scheme is three-level and linear implicit. Existence of its difference solutions has been shown. It is proved by the discrete energy method that the two schemes are uniquely solvable, unconditionally stable and the convergence is of second-order in the maximum norm. An iterative algorithm is proposed for solving the nonlinear scheme. The particular case known as the RLW equation is also discussed numerically in detail. Furthermore, three invariants of motion are evaluated to determine the conservation properties of the problem. Interaction of solitary waves with different amplitudes are shown. The three invariants of the motion are evaluated to determine the conservation proprieties of the system. The temporal evaluation of a Maxwellian initial pulse is then studied. Some numerical examples are given in order to validate the theoretical results.

Suggested Citation

  • Rouatbi, Asma & Omrani, Khaled, 2017. "Two conservative difference schemes for a model of nonlinear dispersive equations," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 516-530.
    • Handle: RePEc:eee:chsofr:v:104:y:2017:i:c:p:516-530
    DOI: 10.1016/j.chaos.2017.09.006
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    Cited by:

    1. Mouktonglang, Thanasak & Yimnet, Suriyon & Sukantamala, Nattakorn & Wongsaijai, Ben, 2022. "Dynamical behaviors of the solution to a periodic initial–boundary value problem of the generalized Rosenau-RLW-Burgers equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 196(C), pages 114-136.
    2. Poochinapan, Kanyuta & Wongsaijai, Ben, 2023. "High-performance computing of structure-preserving algorithm for the coupled BBM system formulated by weighted compact difference operators," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 205(C), pages 439-467.
    3. Teeranush Suebcharoen & Kanyuta Poochinapan & Ben Wongsaijai, 2022. "Bifurcation Analysis and Numerical Study of Wave Solution for Initial-Boundary Value Problem of the KdV-BBM Equation," Mathematics, MDPI, vol. 10(20), pages 1-20, October.
    4. Wongsaijai, Ben & Poochinapan, Kanyuta, 2021. "Optimal decay rates of the dissipative shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation with power of nonlinearity," Applied Mathematics and Computation, Elsevier, vol. 405(C).
    5. Poochinapan, Kanyuta & Wongsaijai, Ben, 2022. "Numerical analysis for solving Allen-Cahn equation in 1D and 2D based on higher-order compact structure-preserving difference scheme," Applied Mathematics and Computation, Elsevier, vol. 434(C).

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