March  2016, 11(1): 1-27. doi: 10.3934/nhm.2016.11.1

A combined finite volume - finite element scheme for a dispersive shallow water system

1. 

Inria, EPC ANGE, 2 rue Simone Iff, F75012 Paris, France, France, France, France

Received  June 2015 Revised  October 2015 Published  January 2016

We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation.
    The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces.
    The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition.
    Several numerical experiments are performed to confirm the relevance of our approach.
Citation: Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
References:
[1]

N. Aïssiouene, M. O. Bristeau, E. Godlewski and J. Sainte-Marie, A robust and stable numerical scheme for a depth-averaged Euler system,, Submitted., ().   Google Scholar

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show all references

References:
[1]

N. Aïssiouene, M. O. Bristeau, E. Godlewski and J. Sainte-Marie, A robust and stable numerical scheme for a depth-averaged Euler system,, Submitted., ().   Google Scholar

[2]

Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4.  Google Scholar

[3]

Indiana Univ. Math. J., 57 (2008), 97-131. doi: 10.1512/iumj.2008.57.3200.  Google Scholar

[4]

SIAM J. Sci. Comput., 25 (2004), 2050-2065. doi: 10.1137/S1064827503431090.  Google Scholar

[5]

E. Audusse, F. Bouchut, M.-O. Bristeau and J. Sainte-Marie, Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system, 2014,, URL , ().   Google Scholar

[6]

Philos. Trans. Royal Soc. London Series A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.  Google Scholar

[7]

European Journal of Mechanics - B/Fluids, 30 (2011), 589-597, URL http://www.sciencedirect.com/science/article/pii/S0997754611000185, Special Issue: Nearshore Hydrodynamics. doi: 10.1016/j.euromechflu.2011.02.005.  Google Scholar

[8]

ESAIM Proc., 15 (2005), 1-17.  Google Scholar

[9]

Birkhäuser, 2004. doi: 10.1007/b93802.  Google Scholar

[10]

Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129-151.  Google Scholar

[11]

Rapport de recherche RR-4282, INRIA, 2001, URL http://hal.inria.fr/inria-00072305, Projet M3N. Google Scholar

[12]

Computers & Fluids, 47 (2011), 51-64. doi: 10.1016/j.compfluid.2011.02.013.  Google Scholar

[13]

Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 961-988, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10801. doi: 10.3934/dcdsb.2015.20.961.  Google Scholar

[14]

Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733-759. doi: 10.3934/dcdsb.2008.10.733.  Google Scholar

[15]

Adv. Appl. Math., 31 (1994), 1-33. doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[16]

J. Sci. Comput., 48 (2011), 105-116. doi: 10.1007/s10915-010-9395-9.  Google Scholar

[17]

Math. Comp., 22 (1968), 745-762. doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[18]

Advanced Series on Ocean Engineering - World Scientific, 1997. doi: 10.1142/9789812796042.  Google Scholar

[19]

Technical Report H1684-12, AST G8M Coastal Morphodybamics Research Programme, 1994. Google Scholar

[20]

Communications in Computational Physics, 17 (2015), 721-760, URL https://hal.archives-ouvertes.fr/hal-00980826. doi: 10.4208/cicp.150414.101014a.  Google Scholar

[21]

SIAM J. Numer. Anal., 32 (1995), 1017-1057. doi: 10.1137/0732047.  Google Scholar

[22]

ESAIM Math. Model. Numer. Anal., 43 (2009), 353-375, URL http://journals.cambridge.org/abstract\_S0764583X05000166. doi: 10.1051/m2an:2008048.  Google Scholar

[23]

Applied Mathematical Sciences, vol. 118, Springer, New York, 1996. doi: 10.1007/978-1-4612-0713-9.  Google Scholar

[24]

J. Fluid Mech., 78 (1976), 237-246. doi: 10.1017/S0022112076002425.  Google Scholar

[25]

P. Gresho and S. Chan, Semi-consistent mass matrix techniques for solving the incompressible Navier-Stokes equations,, First Int. Conf. on Comput. Methods in Flow Analysis, ().   Google Scholar

[26]

ESAIM: Mathematical Modelling and Numerical Analysis, 30 (1996), 637-667, URL http://eudml.org/doc/193818.  Google Scholar

[27]

Math. Comput., 73 (2004), 1719-1737, URL http://dblp.uni-trier.de/db/journals/moc/moc73.html#GuermondS04. doi: 10.1090/S0025-5718-03-01621-1.  Google Scholar

[28]

Journal of Computational Physics, 199 (2004), 221-259, URL http://www.sciencedirect.com/science/article/pii/S002199910400083X. doi: 10.1016/j.jcp.2004.02.009.  Google Scholar

[29]

Physics of Fluids, 21 (2009), 016601. doi: 10.1063/1.3053183.  Google Scholar

[30]

J. Comput. Phys., 229 (2010), 2034-2045. doi: 10.1016/j.jcp.2009.11.021.  Google Scholar

[31]

Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[32]

Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, 119 (1993), 618-638. doi: 10.1061/(ASCE)0733-950X(1993)119:6(618).  Google Scholar

[33]

J. Fluid Mech., 27 (1967), 815-827. doi: 10.1017/S0022112067002605.  Google Scholar

[34]

Masson, 1988. Google Scholar

[35]

in The Navier-Stokes Equations II - Theory and Numerical Methods (eds. G. Heywood John, K. Masuda, R. Rautmann and A. Solonnikov Vsevolod), vol. 1530 of Lecture Notes in Mathematics, Springer Berlin Heidelberg, 1992, 167-183, URL http://dx.doi.org/10.1007/BFb0090341. doi: 10.1007/BFb0090341.  Google Scholar

[36]

J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations,, 11th AIAA Computational Fluid Dynamic Conference, ().   Google Scholar

[37]

SIAM J. Numer. Anal., 32 (1995), 386-403. doi: 10.1137/0732016.  Google Scholar

[38]

J. Fluid Mech., 107 (1981), 499-508. doi: 10.1017/S0022112081001882.  Google Scholar

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