Partial Differential Equations
An analytical solution of the shallow water system coupled to the Exner equation
[Une solution analytique du système de Saint-Venant couplé à lʼéquation dʼExner]
Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 183-186.

Cette Note présente une solution analytique pour le système modélisant le transport de sédiments par le charriage. Cette solution est valable pour une grande famille de lois sédimentaires comme le modèle de Grass ainsi que celui de Meyer-Peter et Müller. Ce résultat est utile pour la validation des schémas numériques.

In this Note, an exact smooth solution for the equations modeling the bedload transport of sediment in shallow water is presented. This solution is valid for a large family of sedimentation laws which are widely used in erosion modeling such as the Grass model or those of Meyer-Peter and Müller. One of the main interests of this solution is the derivation of numerical benchmarks to valid the approximation methods.

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Accepté le :
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DOI : 10.1016/j.crma.2012.01.007
Berthon, Christophe 1 ; Cordier, Stéphane 2 ; Delestre, Olivier 3 ; Le, Minh Hoang 4

1 Laboratoire de mathematiques Jean-Leray, université de Nantes, 2, rue de la Houssiniere, 44322 Nantes, France
2 MAPMO UMR CNRS 6628, université dʼOrléans, UFR sciences, bâtiment de mathématiques, 45067 Orléans, France
3 Laboratoire de mathématiques J.A. Dieudonné & École polytech Nice-Sophia, université de Nice-Sophia Antipolis, parc Valrose, 06108 Nice, France
4 BRGM, 3, avenue Claude-Guillemin, B.P. 36009, 45060 Orléans, France
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Berthon, Christophe; Cordier, Stéphane; Delestre, Olivier; Le, Minh Hoang. An analytical solution of the shallow water system coupled to the Exner equation. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 183-186. doi : 10.1016/j.crma.2012.01.007. http://www.numdam.org/articles/10.1016/j.crma.2012.01.007/

[1] E. Audusse, C. Chalons, O. Delestre, N. Goutal, M. Jodeau, J. Sainte-Marie, B. Spinewine, Sediment transport modelling: Three layer models and relaxation schemes, CEMRACS Proceedings, 2011, submitted for publication.

[2] Castro Díaz, M.J.; Fernández-Nieto, E.D.; Ferreiro, A.M. Sediment transport models in shallow water equations and numerical approach by high order finite volume methods, Computers & Fluids, Volume 37 (March 2008) no. 3, pp. 299-316

[3] Cordier, S.; Le, M.H.; de Luna, T. Morales Bedload transport in shallow water models: Why splitting (may) fail, how hyperbolicity (can) help, Advances in Water Resources, Volume 34 (2011) no. 8, pp. 980-989

[4] O. Delestre, C. Lucas, P.-A. Ksinant, F. Darboux, C. Laguerre, T.N.T. Vo, F. James, S. Cordier, SWASHES: a library of shallow water analytic solutions for hydraulic and environmental studies, submitted for publication, . | arXiv

[5] Exner, F. Über die wechselwirkung zwischen wasser und geschiebe in flüssen, Sitzungsber., Akad. Wissenschaften pt. IIa, Volume 134 (1925)

[6] A.J. Grass, Sediment transport by waves and currents, SERC London Cent. Mar. Technol. Report No. FL29, 1981.

[7] J. Hudson, Numerical technics for morphodynamic modelling, PhD thesis, University of Whiteknights, 2001.

[8] E. Meyer-Peter, R. Müller, Formulas for bed-load transport, in: 2nd meeting IAHSR, Stockholm, Sweden, 1948, pp. 1–26.

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