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

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

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...

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Bibliographic Details
Main Authors: Ampol Duangpan, Ratinan Boonklurb, Lalita Apisornpanich, Phiraphat Sutthimat
Format: Article
Language:English
Published: MDPI AG 2025-08-01
Series:Mathematics
Subjects:
shallow water equations
Chebyshev polynomials
finite integration method
numerical simulation
dam break
fluid dynamics
Online Access:https://www.mdpi.com/2227-7390/13/15/2492
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Summary: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><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>12</mn></mrow></msup></semantics></math></inline-formula> for water height and velocity, while conserving mass with volume deviations under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></semantics></math></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

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