]. A. Afonso, P. J. Oliveira, F. T. Pinho, and M. A. Alves, The log-conformation tensor approach in the finite-volume method framework, Journal of Non-Newtonian Fluid Mechanics, vol.157, issue.1-2, pp.55-65, 2009.
DOI : 10.1016/j.jnnfm.2008.09.007

J. W. Barrett and S. Boyaval, Existence and approximation of a (regularized ) Oldroyd-B model Existence and approximation of a (regularized) Oldroyd-B model, pp.3-21, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00409594

W. John, S. Barrett, and . Boyaval, Finite-element approximation of the FENE-P model, press, working paper arXiv, pp.1704-00886, 2017.

C. Berthon, F. Coquel, and P. G. Lefloch, Why many theories of shock waves are necessary: kinetic relations for non-conservative systems, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.142, issue.01, pp.1-37, 2012.
DOI : 10.1017/S0308210510001009

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Annals of Mathematics, vol.161, issue.1, pp.161-223, 2005.
DOI : 10.4007/annals.2005.161.223
URL : http://annals.math.princeton.edu/wp-content/uploads/annals-v161-n1-p06.pdf

R. , B. Bird, and W. J. Drugan, An exploration and further study of an enhanced oldroyd model, Physics of Fluids, vol.29, issue.5, p.53103, 2017.

F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Mathematics, p.212820965002, 2004.

F. Bouchut and S. Boyaval, A NEW MODEL FOR SHALLOW VISCOELASTIC FLUIDS, Mathematical Models and Methods in Applied Sciences, vol.320, issue.08, pp.3-23, 2013.
DOI : 10.1017/S0022112081001882
URL : https://hal.archives-ouvertes.fr/hal-00628651

F. Bouchut and S. Boyaval, Unified derivation of thin-layer reduced models for shallow free-surface gravity flows of viscous fluids, European Journal of Mechanics - B/Fluids, vol.55, pp.116-131, 2016.
DOI : 10.1016/j.euromechflu.2015.09.003
URL : https://hal.archives-ouvertes.fr/hal-00833468

S. Boyaval, Johnson-Segalman ? Saint-Venant equations for viscoelastic shallow flows in the pure elastic limit, Proceedings in Mathematics and Statistics (PROMS) of the International Conference on Hyperbolic Problems: Theory, Numeric and Applications in Aachen 2016, p.2017

S. Boyaval, C. Tonylelì, and . Mangoubi, Free-energy-dissipative schemes for the Oldroyd-B model, ESAIM: Mathematical Modelling and Numerical Analysis, vol.91, issue.3, pp.523-561, 2009.
DOI : 10.1016/S0377-0257(99)00095-6
URL : https://hal.archives-ouvertes.fr/inria-00204620

G. Chen and D. Wang, The Cauchy Problem for the Euler Equations for Compressible Fluids, Handbook of Mathematical Fluid Dynamics, vol.1, pp.421-543, 2002.
DOI : 10.1016/S1874-5792(02)80012-X

E. Chiodaroli, C. De-lellis, and O. Kreml, Global Ill-Posedness of the Isentropic System of Gas Dynamics, Communications on Pure and Applied Mathematics, vol.45, issue.3, pp.1157-1190, 2015.
DOI : 10.1016/j.crma.2011.09.009

C. M. Dafermos, Hyperbolic conservation laws in continuum physics, 2000.
DOI : 10.1007/978-3-662-22019-1

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

J. Brian, A. N. Edwards, and . Beris, Remarks concerning compressible viscoelastic fluid models, Journal of Non-Newtonian Fluid Mechanics, vol.36, pp.411-417, 1990.

R. Fattal and R. Kupferman, Time-dependent simulation of viscoelastic flows at high Weissenberg number using the log-conformation representation, Journal of Non-Newtonian Fluid Mechanics, vol.126, issue.1, pp.23-27, 2005.
DOI : 10.1016/j.jnnfm.2004.12.003

G. Gallice, Schémas de type godunov entropiques et positifs préservant les discontinuités de contact, Comptes Rendus de l'Académie des Sciences -Series I, Mathematics, vol.331, issue.2, pp.149-152, 2000.
DOI : 10.1016/s0764-4442(00)01601-3

L. Sergey, . Gavrilyuk, and A. Kseniya, Ivanova, and Nicolas Favrie, Multi? dimensional shear shallow water flows : problems and solutions, 2017.

J. Gerbeau and B. Perthame, Derivation of viscous Saint- Venant system for laminar shallow water ; numerical validation, Discrete and continuous dynamical system, Series B, vol.1, issue.1, pp.89-102, 2001.

E. Godlewski and P. Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, vol.118, pp.1410987-98, 1996.
DOI : 10.1007/978-1-4612-0713-9

A. Guaily, E. Cheluget, K. Lee, and M. Epstein, A new hyperbolic model and an experimental study for the flow of polymer melts in Multi-Pass Rheometer, Computers & Fluids, vol.44, issue.1, pp.258-266, 2011.
DOI : 10.1016/j.compfluid.2011.01.007

A. Guaily and M. Epstein, A unified hyperbolic model for viscoelastic liquids, Mechanics Research Communications, vol.37, issue.2, pp.158-163, 2010.
DOI : 10.1016/j.mechrescom.2009.12.004

A. Kseniya, . Ivanova, L. Sergey, B. Gavrilyuk, G. L. Nkonga et al., Formation and coarsening of roll-waves in shear shallow water flows down an inclined rectangular channel, 2016.

M. W. Johnson and D. Segalman, A model for viscoelastic fluid behavior which allows non-affine deformation, Journal of Non-Newtonian Fluid Mechanics, vol.2, issue.3, pp.255-270, 1977.
DOI : 10.1016/0377-0257(77)80003-7

P. Knechtges, M. Behr, and S. Elgeti, Fully-implicit log-conformation formulation of constitutive laws, Journal of Non-Newtonian Fluid Mechanics, vol.214, issue.0, pp.78-87, 2014.
DOI : 10.1016/j.jnnfm.2014.09.018

R. J. Leveque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, p.65001, 2003.
DOI : 10.1017/CBO9780511791253

F. Marche, Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects, European Journal of Mechanics - B/Fluids, vol.26, issue.1, pp.49-63, 2007.
DOI : 10.1016/j.euromechflu.2006.04.007

N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows, Journal de Math??matiques Pures et Appliqu??es, vol.96, issue.5, pp.502-520, 2011.
DOI : 10.1016/j.matpur.2011.04.008

J. Maxwell, On the Dynamical Theory of Gases, Philosophical Transactions of the Royal Society of London, vol.157, issue.0, pp.49-88, 1867.
DOI : 10.1098/rstl.1867.0004

J. G. Oldroyd, On the Formulation of Rheological Equations of State, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.200, issue.1063, pp.523-541, 1950.
DOI : 10.1098/rspa.1950.0035

H. Christian¨ottingerchristian¨ and C. , Stochastic processes in polymeric fluids, p.76008, 1996.

. Tsorng-whay, J. Pan, and . Hao, Numerical simulation of a lid-driven cavity viscoelastic flow at high weissenberg numbers, Comptes Rendus Mathematique, vol.344, issue.4, pp.283-286, 2007.

L. Perrotti, N. J. Walkington, and D. Wang, Numerical approximation of viscoelastic fluids, ESAIM: Mathematical Modelling and Numerical Analysis, vol.45, issue.3, pp.2-51, 2017.
DOI : 10.1051/m2an/2010065

F. R. Phelan, M. F. Malone, and H. H. Winter, A purely hyperbolic model for unsteady viscoelastic flow, Journal of Non-Newtonian Fluid Mechanics, vol.32, issue.2, pp.197-224, 1989.
DOI : 10.1016/0377-0257(89)85036-0

M. Renardy, Mathematical analysis of viscoelastic flows, CBMS-NSF Conference Series in Applied Mathematics, 2000.
DOI : 10.1146/annurev.fl.21.010189.000321

D. Serre, Systèmes de lois de conservation. I, Fondations. [Foundations], Diderot Editeur, Hyperbolicité, entropies, ondes de choc. [Hyperbolicity , entropies, shock waves]. MR 1459988, p.35139, 1996.

R. G. Sousa, R. J. Poole, A. M. Afonso, F. T. Pinho, P. J. Oliveira et al., Lid-driven cavity flow of viscoelastic liquids, Lid-driven cavity flow of viscoelastic liquids, pp.129-138, 2016.
DOI : 10.1016/j.jnnfm.2016.03.001
URL : https://doi.org/10.1016/j.jnnfm.2016.03.001

J. J. Stoker, Water waves: The mathematical theory with applications, 1957.
DOI : 10.1002/9781118033159

C. Ven-te, Open-channel hydraulics, 1959.

V. M. Teshukov, Gas-dynamic analogy for vortex free-boundary flows, Journal of Applied Mechanics and Technical Physics, vol.38, issue.3, pp.303-309, 2007.
DOI : 10.1007/s10808-007-0039-2

V. M. Teshukov, Gas-dynamic analogy in the theory of stratified liquid flows with a free boundary, Fluid Dynamics, vol.42, issue.5, pp.807-817, 2007.
DOI : 10.1134/S0015462807050134