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Numerical dynamics for discrete nonlinear damping Korteweg–de Vries equations

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  • Liu, Guifen
  • Li, Yangrong
  • Wang, Fengling

Abstract

We study the numerical scheme of both solution and attractor for the time-space discrete nonlinear damping Korteweg–de Vries (KdV) equation, which is neither conservative nor coercive. First, we establish a new Taylor expansion as well as a global attractor for the KdV lattice system. Second, we prove the unique existence of numerical solution as well as numerical attractor for the discrete-time KdV lattice system via the implicit Euler scheme. Third, we estimate the discretization error and interpolation error between continuous-time and discrete-time solutions, and then establish the upper semi-convergence from numerical attractors to the global attractor as the time-size tends to zero. Fourth, we establish the finitely dimensional approximation of numerical attractors. Finally, we establish the upper bound as well as the lower semi-convergence of numerical attractors with respect to the external force and the damping constant.

Suggested Citation

  • Liu, Guifen & Li, Yangrong & Wang, Fengling, 2024. "Numerical dynamics for discrete nonlinear damping Korteweg–de Vries equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 225(C), pages 332-349.
    • Handle: RePEc:eee:matcom:v:225:y:2024:i:c:p:332-349
    DOI: 10.1016/j.matcom.2024.05.025
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    References listed on IDEAS

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    1. Kohnesara, Sima Molaei & Firoozjaee, Ali Rahmani, 2023. "Numerical solution of Korteweg–de Vries equation using discrete least squares meshless method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 65-76.
    2. Yang, Shuang & Li, Yangrong, 2022. "Numerical attractors and approximations for stochastic or deterministic sine-Gordon lattice equations," Applied Mathematics and Computation, Elsevier, vol. 413(C).
    3. Ahmad, Fayyaz & Ur Rehman, Shafiq & Zara, Aiman, 2023. "A new approach for the numerical approximation of modified Korteweg–de Vries equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 189-206.
    4. Hoq, Q.E. & Gagnon, J. & Kevrekidis, P.G. & Malomed, B.A. & Frantzeskakis, D.J. & Carretero-González, R., 2009. "Extended nonlinear waves in multidimensional dynamical lattices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 80(4), pages 721-731.
    5. Li, Yangrong & Wang, Fengling & Xia, Huan, 2024. "Continuity-sets of pullback random attractors for discrete porous media equations with colored noise," Applied Mathematics and Computation, Elsevier, vol. 465(C).
    6. Ahmad, Hijaz & Seadawy, Aly R. & Khan, Tufail A., 2020. "Study on numerical solution of dispersive water wave phenomena by using a reliable modification of variational iteration algorithm," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 13-23.
    7. Zhao, Caidi & Jiang, Huite & Caraballo, Tomás, 2021. "Statistical solutions and piecewise Liouville theorem for the impulsive reaction-diffusion equations on infinite lattices," Applied Mathematics and Computation, Elsevier, vol. 404(C).
    8. Abrahamsen, Dylan & Fornberg, Bengt, 2021. "Solving the Korteweg-de Vries equation with Hermite-based finite differences," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    9. Uzunca, Murat & Karasözen, Bülent & Yıldız, Süleyman, 2021. "Structure-preserving reduced-order modeling of Korteweg–de Vries equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 193-211.
    10. Meng, Qingyan & Wang, Yejuan & Kloeden, Peter E., 2023. "The dynamical behavior of a class of stochastic vegetation models," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 211(C), pages 341-367.
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