Commun. Comput. Phys., 7 (2010), pp. 64-102.


The Riemann Problem for the Dispersive Nonlinear Shallow Water Equations

Giovanna Grosso 1*, Matteo Antuono 2, Eleuterio Toro 3

1 DICAT, University of Genova, Via Montallegro 1, I-16145 Genova, Italy.
2 INSEAN (The Italian Ship Model Basin), Via di Vallerano 139, 00128, Roma, Italy.
3 Laboratory of Applied Mathematics, University of Trento, Via Mesiano 77, I-38100 Trento, Italy.

Received 24 November 2008; Accepted (in revised version) 30 March 2009
Available online 16 July 2009
doi:10.4208/cicp.2009.08.181

Abstract

The complete analytical solution of the Riemann problem for the homogeneous Dispersive Nonlinear Shallow Water Equations [Antuono, Liapidevskii and Brocchini, Stud. Appl. Math., 122 (2009), pp. 1-28] is presented, for both wet-bed and dry-bed conditions. Moreover, such a set of hyperbolic and dispersive depth-averaged equations shows an interesting resonance phenomenon in the wave pattern of the solution and we define conditions for the occurrence of resonance and present an algorithm to capture it. As an indirect check on the analytical solution we have carried out a detailed comparison with the numerical solution of the government equations obtained from a dissipative method that does not make explicit use of the solution of the local Riemann problem.


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PACS: 47.11.-j, 02.70.-c, 47.35.-i
Key words: Riemann problem, Dispersive Nonlinear Shallow Water Equations, compound waves, resonance.

*Corresponding author.
Email: ggrosso@dicat.unige.it (G. Grosso), m.antuono@insean.it (M. Antuono), toro@ing.unitn.it (E. Toro)
 

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