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The Construction and Research of the Modified “Upwind Leapfrog” Difference Scheme with Improved Dispersion Properties for the Korteweg–de Vries Equation

Author

Listed:
  • Alexander Sukhinov

    (Department of Mathematics and Informatics, Don State Technical University, 344000 Rostov-on-Don, Russia)

  • Alexander Chistyakov

    (Department of Mathematics and Informatics, Don State Technical University, 344000 Rostov-on-Don, Russia)

  • Elena Timofeeva

    (Department of Computational Mathematics and Cybernetics, North Caucasus Federal University, 355017 Stavropol, Russia)

  • Alla Nikitina

    (Department of Mathematics and Informatics, Don State Technical University, 344000 Rostov-on-Don, Russia)

  • Yulia Belova

    (Department of Mathematics and Informatics, Don State Technical University, 344000 Rostov-on-Don, Russia)

Abstract

This paper covers the construction and research of a scheme to solve the problem with nonlinear dispersion wave equations, described by the model Korteweg–de Vries equation. The article proposes approximating the equation based on improved “Upwind Leapfrog” schemes. Its difference operator is a linear combination of operators of the “Standard Leapfrog” and “Upwind Leapfrog” difference schemes, while the modified scheme is obtained from schemes with optimal weight coefficients. Combining certain values of the weight coefficients mutually compensates for approximation errors. In addition, the modified scheme acquires better properties compared with the original schemes. The results of test calculations of solutions of the nonlinear Korteweg–de Vries equation are presented, illustrating the advantages of the modified scheme.

Suggested Citation

  • Alexander Sukhinov & Alexander Chistyakov & Elena Timofeeva & Alla Nikitina & Yulia Belova, 2022. "The Construction and Research of the Modified “Upwind Leapfrog” Difference Scheme with Improved Dispersion Properties for the Korteweg–de Vries Equation," Mathematics, MDPI, vol. 10(16), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:16:p:2922-:d:887568
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    Citations

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    Cited by:

    1. Kholoud Saad Albalawi & Ibtehal Alazman & Jyoti Geetesh Prasad & Pranay Goswami, 2023. "Analytical Solution of the Local Fractional KdV Equation," Mathematics, MDPI, vol. 11(4), pages 1-13, February.
    2. Alexander Sukhinov & Alexander Chistyakov & Inna Kuznetsova & Yulia Belova & Elena Rahimbaeva, 2022. "Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems," Mathematics, MDPI, vol. 10(19), pages 1-21, September.
    3. Laila F. Seddek & Essam R. El-Zahar & Jae Dong Chung & Nehad Ali Shah, 2023. "A Novel Approach to Solving Fractional-Order Kolmogorov and Rosenau–Hyman Models through the q-Homotopy Analysis Transform Method," Mathematics, MDPI, vol. 11(6), pages 1-11, March.

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