R. Abgrall and S. Karni, Two-Layer Shallow Water System: A Relaxation Approach, SIAM Journal on Scientific Computing, vol.31, issue.3, pp.1603-1627, 2009.
DOI : 10.1137/06067167X

B. Audebert and F. , Structural Stability of Shock Solutions of Hyperbolic Systems in Nonconservation Form via Kinetic Relations, Hyperbolic Problems: Theory, 2008.
DOI : 10.1007/978-3-540-75712-2_36

M. Baer and J. Nunziato, A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials, International Journal of Multiphase Flow, vol.12, issue.6, pp.12-861, 1986.
DOI : 10.1016/0301-9322(86)90033-9

P. G. Baines, Topographic effects in stratified flows, International Journal of Multiphase Flow, vol.22, 1995.
DOI : 10.1016/S0301-9322(97)88112-8

B. Barros, CONSERVATION LAWS FOR ONE-DIMENSIONAL SHALLOW WATER MODELS FOR ONE AND TWO-LAYER FLOWS, Mathematical Models and Methods in Applied Sciences, vol.57, issue.01, pp.119-137, 2006.
DOI : 10.1080/00036819708840534

D. and ?. J. Benney, Some Properties of Long Nonlinear Waves, Studies in Applied Mathematics, vol.45, issue.1, pp.52-97, 1973.
DOI : 10.1002/sapm197352145

C. Berthon, F. Coquel, J. ?. Hérard, and M. Uhlmann, An approximate solution of the Riemann problem for a realisable second-moment turbulent closure, Shock Waves, vol.11, issue.4, pp.11-245, 2002.
DOI : 10.1007/s001930100109

URL : https://hal.archives-ouvertes.fr/hal-01484338

N. Besse, On the Waterbag Continuum, Archive for Rational Mechanics and Analysis, vol.3, issue.8, pp.453-491, 2011.
DOI : 10.1007/s00205-010-0392-9

URL : https://hal.archives-ouvertes.fr/hal-00594878

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources, Birkhäuser, 2004.

R. R. Brock, Development of roll waves in open channels, 1967.

R. R. Brock, Development of roll?wave trains in open channels, J. Hydraulics Division, pp.95-1401, 1969.

R. R. Brock, Periodic permanent roll waves, J. Hydraulics Division, vol.96, pp.2565-2580, 1970.

A. Castro and D. Lannes, Fully nonlinear long-wave models in the presence of??vorticity, Journal of Fluid Mechanics, vol.77, pp.642-675, 2014.
DOI : 10.1029/2002JC001308

F. Coquel, J. ?. Hérard, and K. Saleh, A positive and entropy-satisfying finite volume scheme for the Baer???Nunziato model, Journal of Computational Physics, vol.330, pp.401-435, 2016.
DOI : 10.1016/j.jcp.2016.11.017

URL : https://hal.archives-ouvertes.fr/hal-01261458

G. Dal-maso, P. ?. Lefloch, and F. Murat, Definition and weak stability of a non?conservative product, J. Math. Pures Appli, vol.74, pp.483-548, 1995.

A. N. Dremin and I. A. Karpukhin, Method of determination of shock adiabat of the dispersed substances, Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, vol.1, issue.3, pp.184-188, 1960.

N. Favrie and S. L. Gavrilyuk, Diffuse interface model for compressible fluid-compressible elasticplastic solid interaction, J. Computational Physics, pp.231-2696, 2012.
DOI : 10.1016/j.jcp.2011.11.027

URL : https://hal.archives-ouvertes.fr/hal-01443381

N. Favrie, S. Gavrilyuk, and S. Ndanou, A thermodynamically compatible splitting procedure in hyperelasticity, Journal of Computational Physics, vol.270, pp.300-324, 2014.
DOI : 10.1016/j.jcp.2014.03.051

S. Gavrilyuk and R. Saurel, Estimation of the turbulence energy production across a shock wave, The Journal of Fluid Mechanics, pp.549-131, 2006.

S. L. Gavrilyuk and H. Gouin, Geometric evolution of the Reynolds stress tensor, International Journal of Engineering Science, vol.59, pp.59-65, 2012.
DOI : 10.1016/j.ijengsci.2012.03.008

URL : https://hal.archives-ouvertes.fr/hal-00371444

S. Gavrilyuk, Multiphase flow modelling via Hamilton's principle, In the book, Variational Models And Methods In Solid And Fluid MechanicsSpringer, 2012.

S. K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics, pp.271-306, 1959.

K. A. Ivanova, S. L. Gavrilyuk, B. Nkonga, and G. L. Richard, Formation and coarsening of rollwaves in shear shallow water flows down an inclined rectangular channel
URL : https://hal.archives-ouvertes.fr/hal-01527469

A. K. Kapila, R. Menikoff, J. B. Bdzil, S. F. Son, and D. S. Stewart, Two-phase modeling of deflagration-to-detonation transition in granular materials: Reduced equations, Physics of Fluids, vol.140, issue.10, pp.13-3002, 2001.
DOI : 10.1137/S0036139994268292

S. Karni, Multicomponent Flow Calculations by a Consistent Primitive Algorithm, Journal of Computational Physics, vol.112, issue.1, pp.31-43, 1994.
DOI : 10.1006/jcph.1994.1080

URL : https://deepblue.lib.umich.edu/bitstream/2027.42/31590/1/0000519.pdf

P. and ?. G. Lefloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Applied Mechanics Reviews, vol.56, issue.4, 2002.
DOI : 10.1115/1.1579455

X. Leng and H. Chanson, Breaking bore: Physical observations of roller characteristics, Mechanics Research Communications, vol.65, pp.24-29, 2015.
DOI : 10.1016/j.mechrescom.2015.02.008

URL : http://espace.library.uq.edu.au/view/UQ:354438/Bore337b_highlight.pdf

R. J. Leveque, Numerical methods for conservation laws, 1992.

V. Yu, V. M. Liapidevskii, and . Teshukov, Mathematical models for a long waves propagation in an inhomogeneous fluid, Siberian Branch of the Russian Academy of Sciences, 2000.

K. T. Mandli, Finite volume methods for the multilayer shallow water equations with applications to storm surges: PhD thesis, 2011.

B. Mohammadi and O. Pironneau, Analysis of the K?epsilon turbulence model, Research in Applied Mathematics, 1994.

P. J. Montgomery and T. G. Moodie, On the Number of Conserved Quantities for the Two-Layer Shallow-Water Equations, Studies in Applied Mathematics, vol.106, issue.2, pp.229-259, 2001.
DOI : 10.1111/1467-9590.00166

S. Ndanou, N. Favrie, and S. Gavrilyuk, Multi-solid and multi-fluid diffuse interface model: Applications to dynamic fracture and fragmentation, Journal of Computational Physics, vol.295, pp.295-523, 2015.
DOI : 10.1016/j.jcp.2015.04.024

URL : https://hal.archives-ouvertes.fr/hal-01459183

L. V. 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

L. V. Ovsyannikov, A new solution of the hydrodynamic equations, Dokl. AN SSSR, vol.111, pp.47-49, 1956.

G. L. Richard and S. L. Gavrilyuk, A new model of roll waves: comparison with Brock???s experiments, Journal of Fluid Mechanics, vol.10, pp.698-374, 2012.
DOI : 10.1098/rspa.1984.0079

G. L. Richard and S. L. Gavrilyuk, The classical hydraulic jump in a model of shear shallow-water flows, Journal of Fluid Mechanics, vol.338, pp.725-492, 2013.
DOI : 10.1017/jfm.2012.96

URL : https://hal.archives-ouvertes.fr/hal-01459449

G. L. Richard, Elaboration d'un modèle d'´ ecoulements turbulents en faible profondeur: application au ressaut hydraulique et aux trains de rouleaux, 2013.

R. Saurel and R. , A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows, Journal of Computational Physics, vol.150, issue.2, pp.150-425, 2001.
DOI : 10.1006/jcph.1999.6187

R. Saurel, S. L. Gavrilyuk, and F. Renaud, A multiphase model with internal degrees of freedom: application to shock???bubble interaction, Journal of Fluid Mechanics, vol.495, pp.495-283, 2003.
DOI : 10.1017/S002211200300630X

R. Saurel, O. Le-metayer, J. Massoni, and S. Gavrilyuk, Shock jump relations for multiphase mixtures with stiff mechanical relaxation, Shock Waves, vol.56, issue.3, pp.209-232, 2007.
DOI : 10.1007/s00193-006-0065-7

L. I. Sedov, On the integration of the equations of one-dimensional gas motion, Dokl. AN SSSR, vol.40, pp.753-755, 1953.

V. M. Teshukov, On hyperbolicity of long-wave equations, Soviet Math. Dokl, vol.32, pp.469-473, 1985.

V. M. Teshukov, On Cauchy Problem for Long Wave Equations, pp.331-338, 1992.
DOI : 10.1007/978-3-0348-8627-7_36

V. Teshukov, G. Russo, and A. Chesnokov, ANALYTICAL AND NUMERICAL SOLUTIONS OF THE SHALLOW WATER EQUATIONS FOR 2D ROTATIONAL FLOWS, Mathematical Models and Methods in Applied Sciences, vol.14, issue.10, pp.1451-1479, 2004.
DOI : 10.1002/sapm1983682103

V. M. Teshukov, Gas dynamic analogy for vortex free?boundary flows, Jurnal of Applied Mechanics and Technical Physics, pp.48-303, 2007.

E. F. Toro, Riemann solvers and numerical methods for fluid dynamics: a practical introduction, 2009.
DOI : 10.1007/978-3-662-03490-3

O. V. Troshkin, On wave properties of an incompressible turbulent fluid, Physica A: Statistical Mechanics and its Applications, vol.168, issue.2, pp.881-899, 1990.
DOI : 10.1016/0378-4371(90)90036-R

L. Truskinovsky, Kinks versus Shocks, Shock Induced Transitions and Phase Structures in General Media, 1993.
DOI : 10.1007/978-1-4613-8348-2_11

S. B. Pope, Turbulent flows, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00338511