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A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations

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  • Zhang, Fan
  • Li, Po-Wei
  • Gu, Yan
  • Fan, Chia-Ming

Abstract

In this paper, the space-time generalized finite difference scheme is applied to solve the nonlinear high-order Korteweg-de Vries equations in multiple dimensions. The proposed numerical scheme combines the space-time generalized finite difference method, the Levenberg-Marquardt algorithm, and a time-marching approach. The space-time generalized finite difference method treats the temporal axis as a spatial axis, enabling the proposed scheme to discretize all derivatives in the governing equation. This is accomplished through Taylor series expansion and the moving least squares method. Due to the expandability of the Taylor series to any order, the proposed numerical scheme excels in efficiently handling mixed and higher-order derivatives. These capabilities are distinct advantages of the proposed scheme. The resulting system of algebraic equations is sparse but overdetermined. Therefore, the Levenberg-Marquardt algorithm is directly applied to solve this nonlinear algebraic system. During the calculation process, the time-marching approach reduces computational effort and improves efficiency by dividing the space-time domain.

Suggested Citation

  • Zhang, Fan & Li, Po-Wei & Gu, Yan & Fan, Chia-Ming, 2025. "A space-time generalized finite difference scheme for long wave propagation based on high-order Korteweg-de Vries type equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 228(C), pages 298-312.
    • Handle: RePEc:eee:matcom:v:228:y:2025:i:c:p:298-312
    DOI: 10.1016/j.matcom.2024.09.012
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    References listed on IDEAS

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    1. Qu, Wenzhen & Sun, Linlin & Li, Po-Wei, 2021. "Bending analysis of simply supported and clamped thin elastic plates by using a modified version of the LMFS," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 347-357.
    2. Lin, Ji & Zhang, Yuhui & Reutskiy, Sergiy & Feng, Wenjie, 2021. "A novel meshless space-time backward substitution method and its application to nonhomogeneous advection-diffusion problems," Applied Mathematics and Computation, Elsevier, vol. 398(C).
    3. 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.
    4. 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.
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