M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, American Journal of Physics, vol.34, issue.2, p.20, 1972.
DOI : 10.1119/1.1972842

C. J. Ammon, C. Ji, H. Thio, D. I. Robinson, S. Ni et al., Rupture Process of the 2004 Sumatra-Andaman Earthquake, Science, vol.308, issue.5725, pp.1133-1139, 2005.
DOI : 10.1126/science.1112260

W. Artiles and A. Nachbin, Nonlinear Evolution of Surface Gravity Waves over Highly Variable Depth, Physical Review Letters, vol.93, issue.23, p.234501, 2004.
DOI : 10.1103/PhysRevLett.93.234501

G. A. Athanassoulis and C. E. Papoutsellis, New Form of the Hamiltonian Equations for the Nonlinear Water-Wave Problem, Based on a New Representation of the DTN Operator, and Some Applications, Volume 7: Ocean Engineering, pp.7-13, 2015.
DOI : 10.1115/OMAE2015-41452

T. J. Barth and M. Ohlberger, Finite Volume Methods: Foundation and Analysis, p.17, 2004.
DOI : 10.1002/0470091355.ecm010

J. Basdevant, Variational Principles in Physics, 2007.

S. Beji and K. Nadaoka, A time-dependent nonlinear mild slope equation for water waves, Proc. R. Soc. Lond. A, pp.319-332, 1957.
DOI : 10.1098/rspa.1997.0018

R. C. Berger and G. F. Carey, Free-surface flow over curved surfaces: Part I: Perturbation analysis, International Journal for Numerical Methods in Fluids, vol.128, issue.2, pp.191-200, 1998.
DOI : 10.1002/(SICI)1097-0363(19980815)28:2<191::AID-FLD705>3.0.CO;2-N

P. Bogacki and L. F. Shampine, A 3(2) pair of Runge - Kutta formulas, Applied Mathematics Letters, vol.2, issue.4, pp.321-325, 1989.
DOI : 10.1016/0893-9659(89)90079-7

P. Bonneton, F. Chazel, D. Lannes, F. Marche, and M. Tissier, A splitting approach for the fully nonlinear and weakly dispersive Green???Naghdi model, Journal of Computational Physics, vol.230, issue.4, pp.1479-1498, 2011.
DOI : 10.1016/j.jcp.2010.11.015
URL : https://hal.archives-ouvertes.fr/hal-00482564

F. Bouchut, A. Mangeney-castelnau, B. Perthame, and J. Vilotte, A new model of Saint Venant and Savage???Hutter type for gravity driven shallow water flows, Comptes Rendus Mathematique, vol.336, issue.6, pp.531-536, 2003.
DOI : 10.1016/S1631-073X(03)00117-1

L. J. Broer, On the hamiltonian theory of surface waves, Applied Scientific Research, vol.4, issue.1, pp.430-446, 1974.
DOI : 10.1007/BF00384164

D. Clamond and D. Dutykh, Practical use of variational principles for modeling water waves, Physica D: Nonlinear Phenomena, vol.241, issue.1, pp.25-36, 2012.
DOI : 10.1016/j.physd.2011.09.015
URL : https://hal.archives-ouvertes.fr/hal-00456891

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.

B. J. Dewals, S. Erpicum, P. Archambeau, S. Detrembleur, and M. Pirotton, Depth-integrated flow modelling taking into account bottom curvature, Journal of Hydraulic Research, vol.44, issue.6, pp.785-795, 2006.
DOI : 10.1017/S0022112083000567

R. F. Dressler, NEW NONLINEAR SHALLOW-FLOW EQUATIONS WITH CURVATURE, Journal of Hydraulic Research, vol.49, issue.3, pp.205-222, 1978.
DOI : 10.1080/00221687809499617

D. Dutykh and D. Clamond, Shallow water equations for large bathymetry variations, Journal of Physics A: Mathematical and Theoretical, vol.44, issue.33, p.332001, 2011.
DOI : 10.1088/1751-8113/44/33/332001
URL : https://hal.archives-ouvertes.fr/hal-00580310

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 F. Dias, Water waves generated by a moving bottom, Tsunami and Nonlinear waves, pp.65-96, 2007.
DOI : 10.1007/978-3-540-71256-5_4
URL : https://hal.archives-ouvertes.fr/hal-00115875

D. Dutykh and F. Dias, Energy of tsunami waves generated by bottom motion, Proc. R. Soc. A, pp.725-744, 2009.
DOI : 10.1126/science.1114576
URL : https://hal.archives-ouvertes.fr/hal-00311752

D. Dutykh, F. Dias, and Y. Kervella, Linear theory of wave generation by a moving bottom, Comptes Rendus Mathematique, vol.343, issue.7, pp.499-504, 2006.
DOI : 10.1016/j.crma.2006.09.016
URL : https://hal.archives-ouvertes.fr/hal-00114954

D. Dutykh, T. Katsaounis, and D. Mitsotakis, Finite volume schemes for dispersive wave propagation and runup, Journal of Computational Physics, vol.230, issue.8, pp.3035-3061, 2011.
DOI : 10.1016/j.jcp.2011.01.003
URL : https://hal.archives-ouvertes.fr/hal-00472431

D. Dutykh, T. Katsaounis, and D. Mitsotakis, Finite volume methods for unidirectional dispersive wave models, International Journal for Numerical Methods in Fluids, vol.459, issue.6, pp.717-736, 2013.
DOI : 10.1002/fld.3681
URL : https://hal.archives-ouvertes.fr/hal-00538043

D. Dutykh, C. Labart, and D. Mitsotakis, Long Wave Run-Up on Random Beaches, Physical Review Letters, vol.107, issue.18, pp.184504-184534, 2011.
DOI : 10.1103/PhysRevLett.107.184504

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

D. Dutykh, R. Poncet, and F. Dias, The VOLNA code for the numerical modeling of tsunami waves: Generation, propagation and inundation, European Journal of Mechanics - B/Fluids, vol.30, issue.6, pp.598-615, 2011.
DOI : 10.1016/j.euromechflu.2011.05.005

J. Ghidaglia, A. Kumbaro, and G. L. Coq, On the numerical solution to two fluid models via a cell centered finite volume method, European Journal of Mechanics - B/Fluids, vol.20, issue.6, pp.841-867, 2001.
DOI : 10.1016/S0997-7546(01)01150-5

M. F. Gobbi and J. T. Kirby, Wave evolution over submerged sills: tests of a high-order Boussinesq model, Coastal Engineering, vol.37, issue.1, pp.57-96, 1999.
DOI : 10.1016/S0378-3839(99)00015-0

J. Gray, M. Wieland, and K. Hutter, Gravity-driven free surface flow of granular avalanches over complex basal topography, Proc. R. Soc. Lond. A, pp.1841-1874, 1998.
DOI : 10.1098/rspa.1999.0383

S. T. Grilli, On the Development and Application of Hybrid Numerical Models in Nonlinear Free Surface Hydrodynamics, Proc. 8th Int. Conf. on Hydrodynamics, p.30, 2008.

J. Hammack, A note on tsunamis: their generation and propagation in an ocean of uniform depth, Journal of Fluid Mechanics, vol.51, issue.04, pp.769-799, 1973.
DOI : 10.1103/PhysRev.168.124

A. Harten, ENO schemes with subcell resolution, Journal of Computational Physics, vol.83, issue.1, pp.148-184, 1989.
DOI : 10.1016/0021-9991(89)90226-X

A. Harten and S. Osher, Uniformly High-Order Accurate Nonoscillatory Schemes. I, SIAM Journal on Numerical Analysis, vol.24, issue.2, pp.279-309, 1987.
DOI : 10.1137/0724022
URL : http://www.dtic.mil/get-tr-doc/pdf?AD=ADA158177

J. B. Keller, Shallow-water theory for arbitrary slopes of the bottom, Journal of Fluid Mechanics, vol.489, issue.4, pp.345-348, 2003.
DOI : 10.1017/S0022112003005342

Y. Kervella, D. Dutykh, and F. Dias, Comparison between three-dimensional linear and nonlinear tsunami generation models, Theoretical and Computational Fluid Dynamics, vol.294, issue.4, pp.245-269, 2007.
DOI : 10.1007/s00162-007-0047-0
URL : https://hal.archives-ouvertes.fr/hal-00113909

N. E. Kolgan, Finite-difference schemes for computation of three dimensional solutions of gas dynamics and calculation of a flow over a body under an angle of attack, Uchenye Zapiski TsaGI [Sci. Notes Central Inst. Aerodyn], vol.6, issue.2, pp.1-6, 1975.

R. B. Laughlin, Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Physical Review Letters, vol.50, issue.18, pp.1395-1398, 1983.
DOI : 10.1103/PhysRevLett.50.1395

J. C. Luke, A variational principle for a fluid with a free surface, Journal of Fluid Mechanics, vol.125, issue.02, pp.375-397, 1967.
DOI : 10.1007/BF01449125

P. A. Madsen and H. A. Schaffer, A REVIEW OF BOUSSINESQ-TYPE EQUATIONS FOR SURFACE GRAVITY WAVES, Adv. Coastal & Ocean Engin, vol.5, issue.4, pp.1-94, 1999.
DOI : 10.1142/9789812797544_0001

S. C. Medeiros and S. C. Hagen, Review of wetting and drying algorithms for numerical tidal flow models, International Journal for Numerical Methods in Fluids, vol.57, issue.11, pp.473-487, 2013.
DOI : 10.1002/fld.3668

J. W. Miles and R. Salmon, Weakly dispersive nonlinear gravity waves, Journal of Fluid Mechanics, vol.9, issue.-1, pp.519-531, 1985.
DOI : 10.1016/0378-4371(81)90149-7

A. Nachbin, A Terrain-Following Boussinesq System, SIAM Journal on Applied Mathematics, vol.63, issue.3, pp.905-922, 2003.
DOI : 10.1137/S0036139901397583

K. Nadaoka, S. Beji, and Y. Nakagawa, A fully dispersive weakly nonlinear model for water waves, Proc. R. Soc. Lond. A, pp.303-318, 1957.
DOI : 10.1098/rspa.1997.0017

S. Neetu, I. Suresh, R. Shankar, D. Shankar, S. S. Shenoi et al., Comment on "The Great Sumatra-Andaman Earthquake of 26 December 2004", Science, vol.310, issue.5753, pp.1431-1431, 2004.
DOI : 10.1126/science.1118950

A. A. Petrov, Variational statement of the problem of liquid motion in a container of finite dimensions, Journal of Applied Mathematics and Mechanics, vol.28, issue.4, pp.917-922, 1964.
DOI : 10.1016/0021-8928(64)90077-2

J. J. Sakurai, Modern Quantum Mechanics, 1993.

S. B. Savage and K. Hutter, The motion of a finite mass of granular material down a rough incline, Journal of Fluid Mechanics, vol.196, issue.-1, pp.177-215, 1989.
DOI : 10.1007/BF01180101

L. F. Shampine and M. W. Reichelt, The MATLAB ODE Suite, SIAM Journal on Scientific Computing, vol.18, issue.1, pp.1-22, 1997.
DOI : 10.1137/S1064827594276424
URL : https://hal.archives-ouvertes.fr/hal-01333731

G. Söderlind, Digital filters in adaptive time-stepping, ACM Transactions on Mathematical Software, vol.29, issue.1, pp.1-26, 2003.
DOI : 10.1145/641876.641877

G. Söderlind and L. Wang, Adaptive time-stepping and computational stability, Journal of Computational and Applied Mathematics, vol.185, issue.2, pp.225-243, 2006.
DOI : 10.1016/j.cam.2005.03.008

J. J. Stoker, Water Waves: The mathematical theory with applications. Interscience, p.8, 1957.

J. J. Stoker, Water waves, the mathematical theory with applications, p.13, 1958.

C. E. Synolakis, The runup of long waves, p.28, 1986.

C. E. Synolakis and E. N. Bernard, Tsunami science before and beyond Boxing Day 2004, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.125, issue.5743, pp.2231-2265, 2004.
DOI : 10.1126/science.1114576

V. V. Titov, A. B. Rabinovich, H. O. Mofjeld, R. E. Thomson, and F. I. González, The global reach of the 26, Sumatra tsunami. Science, vol.309, pp.2045-2048, 2004.

M. I. Todorovska and M. D. Trifunac, Generation of tsunamis by a slowly spreading uplift of the sea floor, Soil Dynamics and Earthquake Engineering, vol.21, issue.2, pp.151-167, 2001.
DOI : 10.1016/S0267-7261(00)00096-8

B. Van-leer, Towards the Ultimate Conservative Difference Scheme, Journal of Computational Physics, vol.135, issue.2, pp.101-136, 1979.
DOI : 10.1006/jcph.1997.5704

B. Van-leer, Upwind and High-Resolution Methods for Compressible Flow: From Donor Cell to Residual-Distribution Schemes, 16th AIAA Computational Fluid Dynamics Conference, pp.192-206, 2006.
DOI : 10.2514/6.2003-3559

P. Wessel, Analysis of Observed and Predicted Tsunami Travel Times for the Pacific and Indian Oceans, Pure and Applied Geophysics, vol.166, issue.1-2, pp.301-324, 1928.
DOI : 10.1007/s00024-008-0437-2

G. B. Whitham, A general approach to linear and non-linear dispersive waves using a Lagrangian, Journal of Fluid Mechanics, vol.none, issue.02, pp.273-283, 1965.
DOI : 10.1017/S0022112065000745

T. Y. Wu, A unified theory for modeling water waves, Adv. App. Mech, vol.37, issue.4, pp.1-88, 2001.
DOI : 10.1016/S0065-2156(00)80004-6

Y. Xing and C. Shu, High order finite difference WENO schemes with the exact conservation property for the shallow water equations, Journal of Computational Physics, vol.208, issue.1, pp.206-227, 2005.
DOI : 10.1016/j.jcp.2005.02.006

V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, vol.10, issue.no. 4, pp.190-194, 1968.
DOI : 10.1007/BF00913182