Well-Balanced Second-Order Convex Limiting Technique for Solving the Serre–Green–Naghdi Equations

In this paper, we introduce a numerical method for approximating the dispersive Serre–Green–Naghdi equations with topography using continuous finite elements. The method is an extension of the hyperbolic relaxation technique introduced in Guermond et al. (J Comput Phys 450:110809, 2022). It is expli...

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Bibliographic Details
Published inWater Waves An interdisciplinary journal Vol. 4; no. 3; pp. 409 - 445
Main Authors Guermond, Jean-Luc, Kees, Chris, Popov, Bojan, Tovar, Eric
Format Journal Article
LanguageEnglish
Published Cham Springer International Publishing 01.11.2022
Subjects
Mathematical Applications in the Physical Sciences
Mathematics
Mathematics and Statistics
Original Article
Partial Differential Equations
35L50
Convex limiting
Finite-element method
Well-balanced approximation
Positivity-preserving
Invariant domain
Shallow water
76M10
35L65
Second-order accuracy
Entropy viscosity
Serre
Serre–Green–Naghdi
65M60
65M12
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Summary:In this paper, we introduce a numerical method for approximating the dispersive Serre–Green–Naghdi equations with topography using continuous finite elements. The method is an extension of the hyperbolic relaxation technique introduced in Guermond et al. (J Comput Phys 450:110809, 2022). It is explicit, second-order accurate in space, third-order accurate in time, and is invariant-domain preserving. It is also well balanced and parameter free. Special attention is given to the convex limiting technique when physical source terms are added in the equations. The method is verified with academic benchmarks and validated by comparison with laboratory experimental data.
ISSN:2523-367X
2523-3688
DOI:10.1007/s42286-022-00062-8