N. Bedjaoui and P. G. Lefloch, Diffusive???dispersive travelling waves and kinetic relations V. Singular diffusion and nonlinear dispersion, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, vol.134, issue.05, pp.134815-843, 2004.
DOI : 10.1017/S0308210500003504

URL : http://doi.org/10.1006/jdeq.2000.4009

H. S. Bhat and R. C. Fetecau, A Hamiltonian Regularization of the Burgers Equation, Journal of Nonlinear Science, vol.16, issue.6, pp.615-638, 2006.
DOI : 10.1007/s00332-005-0712-7

H. S. Bhat, R. C. Fetecau, and J. Goodman, A Leray-type regularization for the isentropic Euler equations, Nonlinearity, vol.20, issue.9, pp.2035-2046, 2007.
DOI : 10.1088/0951-7715/20/9/001

S. Bianchini and A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, Annals of Mathematics, vol.161, issue.1, pp.223-342, 2005.
DOI : 10.4007/annals.2005.161.223

URL : http://arxiv.org/abs/math/0111321

J. P. Boyd, The Erc-Log Filter and the Asymptotics of the Euler and Vandeven Sequence Acceleration, Proc. 3rd Int. Conf. Spectral and High Order Methods (ICOSAHOM95), pp.267-276, 1995.

J. M. Burgers, A Mathematical Model Illustrating the Theory of Turbulence, Advances in Applied Mechanics, vol.1, issue.4, pp.171-199, 1948.
DOI : 10.1016/S0065-2156(08)70100-5

R. Camassa, P. Chiu, L. Lee, and T. W. Sheu, Viscous and inviscid regularizations in a class of evolutionary partial differential equations, Journal of Computational Physics, vol.229, issue.19, pp.6676-6687, 2005.
DOI : 10.1016/j.jcp.2010.06.002

G. Chen, Euler Equations and Related Hyperbolic Conservation Laws, Handbook of Differential Equations Evolutionary Equations, pp.1-104, 2005.
DOI : 10.1016/S1874-5717(06)80004-6

D. Clamond, D. Dutykh, and D. Mitsotakis, Conservative modified Serre???Green???Naghdi equations with improved dispersion characteristics, Communications in Nonlinear Science and Numerical Simulation, vol.45, issue.6 8, pp.245-257, 2017.
DOI : 10.1016/j.cnsns.2016.10.009

URL : https://hal.archives-ouvertes.fr/hal-01232370

M. G. Crandall and P. Lions, Viscosity Solutions of Hamilton-Jacobi Equations. Transactions of the, pp.1-42, 1983.
DOI : 10.2307/1999343

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.554.9095

A. J. De-saint-venant, Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l'introduction des marées dans leur lit, C. R. Acad. Sc. Paris, vol.73, issue.4, pp.147-154, 1871.

D. Dutykh and D. Clamond, Modified shallow water equations for significantly varying seabeds, Applied Mathematical Modelling, vol.40, issue.23-24, pp.23-249767, 2016.
DOI : 10.1016/j.apm.2016.06.033

URL : https://hal.archives-ouvertes.fr/hal-00675209

D. Dutykh, D. Clamond, P. Milewski, and D. Mitsotakis, Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, European Journal of Applied Mathematics, vol.9, issue.05, pp.761-787, 2013.
DOI : 10.1017/S0022112065000745

URL : https://hal.archives-ouvertes.fr/hal-00587994

D. Dutykh and D. Mitsotakis, On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete and Continuous Dynamical Systems -Series B, pp.799-818, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00369795

E. Godlewski and P. Raviart, Hyperbolic systems of conservation laws, p.18, 1990.
URL : https://hal.archives-ouvertes.fr/hal-00113734

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

D. Gottlieb and J. S. Hesthaven, Spectral methods for hyperbolic problems, Journal of Computational and Applied Mathematics, vol.128, issue.1-2, pp.83-131, 2001.
DOI : 10.1016/S0377-0427(00)00510-0

URL : http://doi.org/10.1016/s0377-0427(00)00510-0

A. E. Green, N. Laws, and P. M. Naghdi, On the Theory of Water Waves, Proc. R. Soc. Lond. A, pp.43-55, 1974.
DOI : 10.1098/rspa.1974.0072

B. T. Hayes and P. G. Lefloch, Nonclassical Shocks and Kinetic Relations: Strictly Hyperbolic Systems, SIAM Journal on Mathematical Analysis, vol.31, issue.5, pp.941-991, 2000.
DOI : 10.1137/S0036141097319826

H. Holden and N. H. Risebro, Riemann Problems with a Kink, SIAM Journal on Mathematical Analysis, vol.30, issue.3, pp.497-515, 1999.
DOI : 10.1137/S0036141097327033

J. K. Hunter and Y. Zheng, On a nonlinear hyperbolic variational equation: II. The zero-viscosity and dispersion limits, Archive for Rational Mechanics and Analysis, vol.45, issue.4, pp.355-383, 1995.
DOI : 10.1007/BF00379260

C. I. Kondo and P. G. Lefloch, Zero Diffusion-Dispersion Limits for Scalar Conservation Laws, SIAM Journal on Mathematical Analysis, vol.33, issue.6, pp.1320-1329, 2002.
DOI : 10.1137/S0036141000374269

URL : http://arxiv.org/abs/0712.0094

P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, issue.4, 1973.

P. D. Lax and C. D. Levermore, The small dispersion limit of the KdV equations: III, Commun. Pure Appl. Math, vol.5, pp.809-830, 1983.

J. Leray, Essai sur les mouvements plans d'un fluide visqueux que limitent des parois, J. Math. Pures Appl, vol.13, issue.5, pp.331-418, 1934.

A. J. Majda and A. L. Bertozzi, Vorticity and Incompressible Flow, 2001.
DOI : 10.1017/CBO9780511613203

V. F. Nesterenko, Dynamics of Heterogeneous Materials, 2001.
DOI : 10.1007/978-1-4757-3524-6

G. Norgard and K. Mohseni, A New Potential Regularization of the One-Dimensional Euler and Homentropic Euler Equations, Multiscale Modeling & Simulation, vol.8, issue.4, pp.1212-1243, 2005.
DOI : 10.1137/090763640

G. Norgard and K. Mohseni, An examination of the homentropic Euler equations with averaged characteristics, Journal of Differential Equations, vol.248, issue.3, pp.574-593, 2005.
DOI : 10.1016/j.jde.2009.08.019

F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux, pp.374-388, 1953.

C. H. Su and C. S. Gardner, Korteweg???de Vries Equation and Generalizations. III. Derivation of the Korteweg???de Vries Equation and Burgers Equation, Journal of Mathematical Physics, vol.10, issue.3, pp.536-539, 1969.
DOI : 10.1103/PhysRevLett.17.996

J. Von-neumann and R. D. Richtmyer, A Method for the Numerical Calculation of Hydrodynamic Shocks, Journal of Applied Physics, vol.21, issue.3, p.232, 1950.
DOI : 10.1007/BF01448839

G. Wei, J. T. Kirby, S. T. Grilli, and R. Subramanya, A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves, Journal of Fluid Mechanics, vol.107, issue.-1, pp.71-92, 1995.
DOI : 10.1063/1.865459

T. Y. Wu, . Unified, . For, D. Water, J. A. Clamond et al., Parc Valrose, F-06108 Nice cedex 2, France E-mail address: diderc@unice.fr URL: http://math.unice.fr, Adv. App. Mech, vol.37, issue.7351, pp.1-88, 2001.