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Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

Author

Listed:
  • Alexander Sukhinov

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

  • Alexander Chistyakov

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

  • Inna Kuznetsova

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

  • Yulia Belova

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

  • Elena Rahimbaeva

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

Abstract

Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:19:p:3564-:d:929544
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    References listed on IDEAS

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    1. Jürgen Geiser & Jose L. Hueso & Eulalia Martínez, 2020. "Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations," Mathematics, MDPI, vol. 8(3), pages 1-22, February.
    2. 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.
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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. 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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    2. 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.

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