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

Ampol Duangpan, Ratinan Boonklurb (), Lalita Apisornpanich and Phiraphat Sutthimat ()
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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
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

Mathematics, 2025, vol. 13, issue 15, 1-30

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.

Keywords: shallow water equations; Chebyshev polynomials; finite integration method; numerical simulation; dam break; fluid dynamics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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