E. Audusse, A multilayer Saint-Venant model: Derivation and numerical validation, Discrete and Continuous Dynamical Systems - Series B, vol.5, issue.2, pp.189-214, 2005.
DOI : 10.3934/dcdsb.2005.5.189

E. Audusse, F. Bouchut, M. Bristeau, R. Klein, and B. Perthame, A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows, SIAM Journal on Scientific Computing, vol.25, issue.6, pp.6-2050, 2004.
DOI : 10.1137/S1064827503431090

A. Bedford and D. S. Drumheller, Theories of immiscible and structured mixtures, International Journal of Engineering Science, vol.21, issue.8, pp.863-960, 1983.
DOI : 10.1016/0020-7225(83)90071-X

A. Bermudez and M. E. Vazquez, Upwind methods for hyperbolic conservation laws with source terms, Computers & Fluids, vol.23, issue.8, pp.1049-1071, 1994.
DOI : 10.1016/0045-7930(94)90004-3

M. Bernard, S. Dellacherie, G. Faccanoni, B. Grec, and Y. Penel, Study of a low Mach nuclear core model for two-phase flows with phase transition I: stiffened gas law, ESAIM: Mathematical Modelling and Numerical Analysis, vol.48, issue.6, pp.48-1639, 2014.
DOI : 10.1051/m2an/2014015
URL : https://hal.archives-ouvertes.fr/hal-00747616

S. Bianchini, On the riemann problem for non-conservative hyperbolic systems, Arch. Ration. Mech. Anal, vol.166, issue.1, pp.1-26, 2003.

F. Bouchut and T. M. De-luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment, ESAIM: Mathematical Modelling and Numerical Analysis, vol.42, issue.4, pp.683-698, 2008.
DOI : 10.1051/m2an:2008019

M. Castro, J. M. Gallardo, J. A. López-garcía, and C. Parés, Well-Balanced High Order Extensions of Godunov's Method for Semilinear Balance Laws, SIAM Journal on Numerical Analysis, vol.46, issue.2, pp.1012-1039, 2008.
DOI : 10.1137/060674879

M. J. Castro, E. D. Fernández-nieto, A. M. Ferreiro, J. A. García-rodríguez, and C. Parés, High Order Extensions of Roe Schemes for??Two-Dimensional Nonconservative Hyperbolic Systems, Journal of Scientific Computing, vol.208, issue.2, pp.1-67, 2009.
DOI : 10.1007/s10915-008-9250-4

C. Chainais-hillairet, Y. Peng, and I. Violet, Numerical solutions of Euler???Poisson systems for potential flows, Applied Numerical Mathematics, vol.59, issue.2, pp.301-315, 2009.
DOI : 10.1016/j.apnum.2008.02.006
URL : https://hal.archives-ouvertes.fr/hal-00489214

A. B. De-saint-venant, Théorie du mouvement non permanent des eaux, avec application aux crues desrivì eres etàetà l'introduction des marées dans leurs lits, C.R. Acad. Sci. Paris, vol.73, pp.147-154, 1871.

S. Dellacherie, Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number, Journal of Computational Physics, vol.229, issue.4, pp.978-1016, 2010.
DOI : 10.1016/j.jcp.2009.09.044

V. Duchêne, Asymptotic Shallow Water Models for Internal Waves in a Two-Fluid System with a Free Surface, SIAM Journal on Mathematical Analysis, vol.42, issue.5, pp.2229-2260, 2010.
DOI : 10.1137/090761100

W. Gangbo and M. Westdickenberg, Optimal Transport for the System of Isentropic Euler Equations, Communications in Partial Differential Equations, vol.58, issue.9, pp.1041-1073, 2009.
DOI : 10.1090/S0002-9947-1951-0042035-0

S. Gavrilyuk and H. Gouin, A new form of governing equations of fluids arising from Hamilton's principle, International Journal of Engineering Science, vol.37, issue.12, pp.12-1495, 1999.
DOI : 10.1016/S0020-7225(98)00131-1

E. Godlewski, R. , and P. , Numerical approximation of hyperbolic systems of conservation laws, of Applied Mathematical Sciences, 1996.
DOI : 10.1007/978-1-4612-0713-9

J. M. Greenberg and A. Leroux, A Well-Balanced Scheme for the Numerical Processing of Source Terms in Hyperbolic Equations, SIAM Journal on Numerical Analysis, vol.33, issue.1, pp.1-16, 1996.
DOI : 10.1137/0733001

N. Grenier, J. Vila, and P. Villedieu, An accurate low-Mach scheme for a compressible two-fluid model applied to free-surface flows, Journal of Computational Physics, vol.252, pp.1-19, 2013.
DOI : 10.1016/j.jcp.2013.06.008

H. Guillard and C. Viozat, On the behaviour of upwind schemes in the low Mach number limit, Computers & Fluids, vol.28, issue.1, pp.63-86, 1999.
DOI : 10.1016/S0045-7930(98)00017-6
URL : https://hal.archives-ouvertes.fr/hal-00871725

M. Ilcak, A. J. Adcroft, S. M. Griffies, and R. W. Hallberg, Spurious dianeutral mixing and the role of momentum closure. Ocean Modelling, pp.45-46, 2012.

S. Jin, Efficient Asymptotic-Preserving (AP) Schemes For Some Multiscale Kinetic Equations, SIAM Journal on Scientific Computing, vol.21, issue.2, pp.441-454, 1999.
DOI : 10.1137/S1064827598334599

S. Jin, W. , and X. , Two Interface-Type Numerical Methods for Computing Hyperbolic Systems with Geometrical Source Terms Having Concentrations, SIAM Journal on Scientific Computing, vol.26, issue.6, pp.2079-2101, 2005.
DOI : 10.1137/040605825

S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Communications on Pure and Applied Mathematics, vol.71, issue.4, pp.481-524, 1981.
DOI : 10.1002/cpa.3160340405

S. Klainerman and A. Majda, Compressible and incompressible fluids, Communications on Pure and Applied Mathematics, vol.33, issue.5, pp.629-651, 1982.
DOI : 10.1002/cpa.3160350503

P. Lax and B. Wendroff, Systems of conservation laws, Communications on Pure and Applied Mathematics, vol.47, issue.2, pp.217-237, 1960.
DOI : 10.1002/cpa.3160130205

M. Liou, S. Jr, and C. J. , A New Flux Splitting Scheme, Journal of Computational Physics, vol.107, issue.1, pp.23-39, 1993.
DOI : 10.1006/jcph.1993.1122

R. Liska and B. Wendroff, Analysis and Computation with Stratified Fluid Models, Journal of Computational Physics, vol.137, issue.1, pp.212-244, 1997.
DOI : 10.1006/jcph.1997.5806

R. R. Long, LONG WAVES IN A TWO-FLUID SYSTEM, Journal of Meteorology, vol.13, issue.1, pp.70-74, 1956.
DOI : 10.1175/1520-0469(1956)013<0070:LWIATF>2.0.CO;2

G. Métivier and S. Schochet, The Incompressible Limit of the Non-Isentropic Euler Equations, Archive for Rational Mechanics and Analysis, vol.158, issue.1, pp.61-90, 2001.
DOI : 10.1007/PL00004241

R. Monjarret, Local well-posedness of the multi-layer shallow water model with free surface

R. Monjarret, Local Well-Posedness of the Two-Layer Shallow Water Model with Free Surface, SIAM Journal on Applied Mathematics, vol.75, issue.5, pp.2311-2332, 2015.
DOI : 10.1137/140957020

M. L. Muñoz-ruiz and C. Parés, Some comments on the numerical approximation of hyperbolic nonconservative systems. In Numerical methods for hyperbolic equations: theory and applications, Balkema, pp.311-318

R. Natalini, M. Ribot, and M. Twarogowska, A well-balanced numerical scheme for a one-dimensional quasilinear hyperbolic model of chemotaxis, Communications in Mathematical Sciences, vol.12, issue.1, pp.13-39, 2014.
DOI : 10.4310/CMS.2014.v12.n1.a2
URL : https://hal.archives-ouvertes.fr/hal-00764086

S. Noelle, N. Pankratz, G. Puppo, and J. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, Journal of Computational Physics, vol.213, issue.2, pp.474-499, 2006.
DOI : 10.1016/j.jcp.2005.08.019

S. Noelle, Y. Xing, and C. Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, Journal of Computational Physics, vol.226, issue.1, pp.29-58, 2007.
DOI : 10.1016/j.jcp.2007.03.031

L. Ovsyannikov, Two-layer ?Shallow water? model, Journal of Applied Mechanics and Technical Physics, vol.2, issue.2, pp.127-135, 1979.
DOI : 10.1007/BF00910010

M. Parisot and J. Vila, Numerical scheme for multilayer shallow-water model in the low-Froude number regime, Comptes Rendus Mathematique, vol.352, issue.11, pp.953-957, 2014.
DOI : 10.1016/j.crma.2014.09.020
URL : https://hal.archives-ouvertes.fr/hal-01079843

B. Perthame and C. Simeoni, A kinetic scheme for the Saint-Venant system??with a source term, Calcolo, vol.38, issue.4, pp.201-231, 2001.
DOI : 10.1007/s10092-001-8181-3
URL : https://hal.archives-ouvertes.fr/hal-00922664

R. Rannacher, On chorin's projection method for the incompressible navier-stokes equations, Lecture Notes in Math, vol.33, pp.167-183, 1991.
DOI : 10.1016/0045-7825(86)90025-3

G. Reznik, V. Zeitlin, B. Jelloul, and M. , Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model, Journal of Fluid Mechanics, vol.445, pp.93-120, 2001.
DOI : 10.1017/S002211200100550X
URL : https://hal.archives-ouvertes.fr/hal-00310857

J. Serrin, Mathematical Principles of Classical Fluid Mechanics, Handbuch der Physik (herausgegeben von S. Flügge), Bd
DOI : 10.1007/978-3-642-45914-6_2

M. Slemrod, Dynamic phase transitions in a van der Waals fluid, Journal of Differential Equations, vol.52, issue.1, pp.1-23, 1984.
DOI : 10.1016/0022-0396(84)90130-X

W. A. Strauss, Partial Differential Equations: An Introduction, 1992.

L. Wijngaarden, On the equations of motion for mixtures of liquid and gas bubbles, Journal of Fluid Mechanics, vol.39, issue.03, pp.3-465, 1968.
DOI : 10.1063/1.1706447