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Finite Integration Method with Chebyshev Expansion for Shallow Water Equations over Variable Topography

Author

Listed:
  • Ampol Duangpan

    (Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand)

  • Ratinan Boonklurb

    (Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
    These authors contributed equally to this work.)

  • Lalita Apisornpanich

    (Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand)

  • Phiraphat Sutthimat

    (Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand
    These authors contributed equally to this work.)

Abstract

The shallow water equations (SWEs) model fluid flow in rivers, coasts, and tsunamis. Their nonlinearity challenges analytical solutions. We present a numerical algorithm combining the finite integration method with Chebyshev polynomial expansion (FIM-CPE) to solve one- and two-dimensional SWEs. The method transforms partial differential equations into integral equations, approximates spatial terms via Chebyshev polynomials, and uses forward differences for time discretization. Validated on stationary lakes, dam breaks, and Gaussian pulses, the scheme achieved errors below 10 − 12 for water height and velocity, while conserving mass with volume deviations under 10 − 5 . Comparisons showed superior shock-capturing versus finite difference methods. For two-dimensional cases, it accurately resolved wave interactions over complex topographies. Though limited to wet beds and small-scale two-dimensional problems, the method provides a robust simulation tool.

Suggested Citation

  • Ampol Duangpan & Ratinan Boonklurb & Lalita Apisornpanich & Phiraphat Sutthimat, 2025. "Finite Integration Method with Chebyshev Expansion for Shallow Water Equations over Variable Topography," Mathematics, MDPI, vol. 13(15), pages 1-30, August.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:15:p:2492-:d:1716260
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    References listed on IDEAS

    as
    1. Szu-Hsien Peng, 2012. "1D and 2D Numerical Modeling for Solving Dam-Break Flow Problems Using Finite Volume Method," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-14, April.
    2. Huamin Zhang & Feng Ding, 2013. "On the Kronecker Products and Their Applications," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    3. Huamin Zhang & Feng Ding, 2013. "On the Kronecker Products and Their Applications," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-8, June.
    4. Szu-Hsien Peng, 2012. "1D and 2D Numerical Modeling for Solving Dam‐Break Flow Problems Using Finite Volume Method," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
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