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A fast finite volume solver for multi-layered shallow water flows with mass exchange

Audusse, E.; Benkhaldoun, B.; Sari, S.; Seaid, M.; Tassi, P.

A fast finite volume solver for multi-layered shallow water flows with mass exchange Thumbnail


E. Audusse

B. Benkhaldoun

S. Sari

P. Tassi


A fast finite volume solver for hydrostatic multi-layered shallow water flows with mass exchange is investigated. In contrast to many models for multi-layered hydrostatic shallow water flows where the immiscible suppression is assumed, the present model allows for mass exchange between the layers. The multi-layered shallow water equations form a system of conservation laws with source terms for which the computation of the eigenvalues is not trivial. For most practical applications, complex eigenvalues may arise in the system and the multi-layered shallow water equations are not hyperbolic anymore. This property makes the application of conventional finite volume methods difficult or even impossible for those methods that require in their formulation the explicit computation of the eigenvalues. In the current study, we propose a finite volume method that avoids the solution of Riemann problems. At each time step, the method consists of two stages to update the new solution. In the first stage, the multi-layered shallow water equations are rewritten in a non-conservative form and the intermediate solutions are calculated using the method of characteristics. In the second stage, the numerical fluxes are reconstructed from the intermediate solutions in the first stage and used in the conservative form of the multi-layered shallow water equations. The proposed method is simple to implement, satisfies the conservation property and is suitable for multi-layered shallow water equations on non-flat topography. The proposed finite volume solver is verified against several benchmark tests and it shows good agreement with analytical solutions of the incompressible hydrostatic Navier–Stokes equations. The method is conservative by construction and preserves the mass to the machine precision. The performance of the method is also demonstrated by comparing the results obtained using the proposed finite volume method to those obtained using the well-established kinetic method.


Audusse, E., Benkhaldoun, B., Sari, S., Seaid, M., & Tassi, P. (2014). A fast finite volume solver for multi-layered shallow water flows with mass exchange. Journal of Computational Physics, 272, 23-45.

Journal Article Type Article
Acceptance Date Apr 9, 2014
Online Publication Date Apr 16, 2014
Publication Date Sep 1, 2014
Deposit Date Jun 2, 2014
Publicly Available Date Jun 3, 2014
Journal Journal of Computational Physics
Print ISSN 0021-9991
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 272
Pages 23-45
Keywords Multi-layer shallow water equations, Hydrostatic flows, Finite volume method, Method of characteristics, Mass exchange, Wind-driven flows.


Accepted Journal Article (4.4 Mb)

Copyright Statement
NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational Physics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational Physics, 272, 2014, 10.1016/

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