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

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Publicado en:Mathematics vol. 13, no. 15 (2025), p. 2492-2522
Autor principal: Ampol, Duangpan
Otros Autores: Ratinan, Boonklurb, Apisornpanich Lalita, Phiraphat, Sutthimat
Publicado:
MDPI AG
Materias:
Finite volume method
Velocity
Partial differential equations
Shallow water equations
Topography
Finite difference method
Integral equations
Numerical analysis
Polynomials
Fluid flow
Water
Chebyshev approximation
Exact solutions
Approximation
Wave interaction
Acceso en línea:Citation/Abstract
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Resumen: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 <inline-formula>10−12</inline-formula> for water height and velocity, while conserving mass with volume deviations under <inline-formula>10−5</inline-formula>. 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.
ISSN:2227-7390
DOI:10.3390/math13152492
Fuente:Engineering Database