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Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method

Author

Listed:
  • Huimin Wang

    (College of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, China)

  • Yanhong Liu

    (College of Mathematics, Jilin University, Changchun 130012, China)

  • Xiuling Li

    (College of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, China)

  • Hengjia Chen

    (College of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, China)

Abstract

The generalized Zakharov equation is a widely used and crucial model in plasma physics, which helps to understand wave particle interactions and nonlinear wave propagation in plasma. The solitary wave solution of this equation provides insights into phenomena such as electron and ion acoustic waves, as well as magnetic field disturbances in plasma. The numerical simulation of solitary wave solutions to the generalized Zakharov equation is an interesting problem worth studying. This is crucial for plasma-based technology, as well as for understanding nonlinear wave propagation in plasma physics and other fields. In this study, a numerical investigation of the generalized Zakharov equation using the lattice Boltzmann method has been conducted. The lattice Boltzmann method is a new modeling and simulating method at the mesoscale. A lattice Boltzmann model was constructed by performing Taylor expansion, Chapman–Enskog expansion, and time multiscale expansion on the lattice Boltzmann equation. By defining the moments of the equilibrium distribution function appropriately, the macroscopic equation can be restored. Furthermore, the numerical experiments for the equation are carried out with the parameter lattice size m = 100 , time step Δ t = 0.001 , and space step size Δ x = 0.4 . The solitary wave solution of the equation is numerically simulated. Numerical results under different parameter values are compared, and the convergence and effectiveness of the model are numerically verified. It is obtained that the model is convergent in time and space, and the convergence orders are all 2.24881. The effectiveness of our model was also verified by comparing the numerical results of different numerical methods. The lattice Boltzmann method demonstrates advantages in both accuracy and CPU time. The results indicate that the lattice Boltzmann method is a good tool for computing the generalized Zakharov equation.

Suggested Citation

  • Huimin Wang & Yanhong Liu & Xiuling Li & Hengjia Chen, 2024. "Numerical Simulation for Solitary Waves of the Generalized Zakharov Equation Based on the Lattice Boltzmann Method," Mathematics, MDPI, vol. 12(7), pages 1-14, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:967-:d:1363279
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    References listed on IDEAS

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    1. Wang, Huimin, 2016. "A lattice Boltzmann model for the ion- and electron-acoustic solitary waves in beam-plasma system," Applied Mathematics and Computation, Elsevier, vol. 279(C), pages 62-75.
    2. Lai, Huilin & Ma, Changfeng, 2014. "A new lattice Boltzmann model for solving the coupled viscous Burgers’ equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 445-457.
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