M. Antuono, V. Y. Liapidevskii, and M. Brocchini, Dispersive Nonlinear Shallow-Water Equations, Studies in Applied Mathematics, vol.15, issue.1, pp.1-28, 2009.
DOI : 10.1111/j.1467-9590.2008.00422.x

E. Audusse, Modélisation hyperbolique et analyse numérique pour les écoulements en eaux peu profondes, p.19, 2004.

T. J. Barth, Aspects of unstructured grids and finite-volume solvers for the Euler and Navier- Stokes equations. Lecture series -van Karman Institute for Fluid Dynamics, pp.1-140, 1994.

T. J. Barth and M. Ohlberger, Finite Volume Methods: Foundation and Analysis, Encyclopedia of Computational Mechanics, 1921.
DOI : 10.1002/9781119176817.ecm2010

URL : http://hdl.handle.net/2060/20030020790

G. K. Batchelor, An introduction to fluid dynamics, volume 61 of Cambridge mathematical library, p.29, 2000.

S. A. Beisel, L. B. Chubarov, D. Dutykh, G. S. Khakimzyanov, and N. Y. Shokina, Simulation of surface waves generated by an underwater landslide in a bounded reservoir, Russian Journal of Numerical Analysis and Mathematical Modelling, vol.27, issue.6, pp.539-558
DOI : 10.1515/rnam-2012-0031

S. Bellec and M. Colin, On the existence of solitary waves for Boussinesq type equations and Cauchy problem for a new conservative model, Adv. Differential Equations, vol.2110, issue.9 5, pp.945-976, 2016.

G. Bellotti and M. Brocchini, On the shoreline boundary conditions for Boussinesq-type models, International Journal for Numerical Methods in Fluids, vol.5, issue.4, pp.479-500, 2001.
DOI : 10.5194/npg-5-27-1998

G. Bellotti and M. Brocchini, On using Boussinesq-type equations near the shoreline: a note of caution, Ocean Engineering, vol.29, issue.12, pp.1569-1575, 2002.
DOI : 10.1016/S0029-8018(01)00092-0

T. B. Benjamin and P. J. Olver, Hamiltonian structure, symmetries and conservation laws for water waves, Journal of Fluid Mechanics, vol.32, issue.-1, pp.137-185, 1982.
DOI : 10.1080/14786449108621390

F. Benkhaldoun and M. Seaïd, New finite-volume relaxation methods for the third-order differential equations, Commun. Comput. Phys, vol.4, issue.4, pp.820-837, 2008.

E. N. Bernard and V. V. Titov, Improving Tsunami Forecast Skill Using Deep Ocean Observations, Marine Technology Society Journal, vol.40, issue.4, pp.23-26, 2007.
DOI : 10.4031/002533206787353223

G. W. Bluman, A. F. Cheviakov, and S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, 2010.
DOI : 10.1007/978-0-387-68028-6

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

J. V. Boussinesq, Théorie de l'intumescence liquide appelée onde solitaire ou de translation se propageant dans un canal rectangulaire, C.R. Acad. Sci. Paris Sér. A-B, vol.72, issue.4, pp.755-759, 1871.

J. P. Boyd, Solitons from sine waves: Analytical and numerical methods for non-integrable solitary and cnoidal waves, Physica D: Nonlinear Phenomena, vol.21, issue.2-3, pp.227-246, 1986.
DOI : 10.1016/0167-2789(86)90003-5

J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2000.
DOI : 10.1007/978-3-642-83876-7

J. P. Boyd, A Comparison of Numerical Algorithms for Fourier Extension of the First, Second, and Third Kinds, Journal of Computational Physics, vol.178, issue.1, pp.118-160, 2002.
DOI : 10.1006/jcph.2002.7023

J. P. Boyd, Deleted Residuals, the QR-Factored Newton Iteration, and Other Methods for Formally Overdetermined Determinate Discretizations of Nonlinear Eigenproblems for Solitary, Cnoidal, and Shock Waves, Journal of Computational Physics, vol.179, issue.1, pp.216-237, 2002.
DOI : 10.1006/jcph.2002.7052

M. Brocchini, A reasoned overview on Boussinesq-type models: the interplay between physics, mathematics and numerics, Proc. R. Soc. A, p.46920130496, 2013.
DOI : 10.1007/s10652-012-9252-5

J. Chambarel, C. Kharif, and J. Touboul, Head-on collision of two solitary waves and residual falling jet formation, Nonlinear Processes in Geophysics, vol.16, issue.1, pp.111-122, 2009.
DOI : 10.5194/npg-16-111-2009

URL : https://hal.archives-ouvertes.fr/hal-00703724

A. F. Cheviakov, Computation of fluxes of conservation laws, Journal of Engineering Mathematics, vol.5, issue.2, pp.153-173, 2010.
DOI : 10.1017/S0956792501004715

C. I. Christov, An energy-consistent dispersive shallow-water model, Wave Motion, vol.34, issue.2, pp.161-174, 2001.
DOI : 10.1016/S0165-2125(00)00082-2

URL : http://www.ucs.louisiana.edu/~cic6380/Articles/WaveMoti.pdf.gz

L. B. Chubarov, G. S. Khakimzyanov, and N. Y. Shokina, Numerical Modelling of Surface Water Waves Arising Due to Movement of Underwater Landslide on Irregular Bottom Slope, Notes on Numerical Fluid Mechanics and Multidisciplinary Design: Computational Science and High Performance Computing IV, pp.75-91, 2011.
DOI : 10.1007/978-3-642-17770-5_7

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

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, pp.245-257, 2017.
DOI : 10.1016/j.cnsns.2016.10.009

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

A. I. Delis, M. Kazolea, and N. A. Kampanis, A robust high-resolution finite volume scheme for the simulation of long waves over complex domains, International Journal for Numerical Methods in Fluids, vol.29, issue.4, pp.419-452, 2008.
DOI : 10.1017/CBO9780511791253

M. D. Risio, G. Bellotti, A. Panizzo, and P. Girolamo, Three-dimensional experiments on landslide generated waves at a sloping coast, Coastal Engineering, vol.56, issue.5-6, pp.659-671, 1926.
DOI : 10.1016/j.coastaleng.2009.01.009

V. A. Dougalis, A. Durán, M. A. Lopez-marcos, and D. E. Mitsotakis, A Numerical Study of the Stability of Solitary Waves of??the??Bona???Smith Family of Boussinesq Systems, Journal of Nonlinear Science, vol.15, issue.6, pp.595-607, 2007.
DOI : 10.1090/conm/200/02517

A. Duran, D. Dutykh, and D. Mitsotakis, On the Galilean Invariance of Some Nonlinear Dispersive Wave Equations, Studies in Applied Mathematics, vol.30, issue.5, pp.359-388, 2013.
DOI : 10.1016/j.apnum.2009.03.002

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, Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting, Mathematics and Computers in Simulation, vol.80, issue.4, pp.837-848, 2009.
DOI : 10.1016/j.matcom.2009.08.036

URL : https://hal.archives-ouvertes.fr/hal-00174439

D. Dutykh and F. Dias, Influence of sedimentary layering on tsunami generation, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.21-22, pp.21-221268, 2010.
DOI : 10.1016/j.cma.2009.07.011

URL : https://hal.archives-ouvertes.fr/hal-00288696

D. Dutykh and H. Kalisch, Boussinesq modeling of surface waves due to underwater landslides, Nonlinear Processes in Geophysics, vol.20, issue.3, pp.267-285, 1926.
DOI : 10.5194/npg-20-267-2013

URL : https://hal.archives-ouvertes.fr/hal-00654386

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, 1920.
DOI : 10.1016/j.jcp.2011.01.003

URL : https://hal.archives-ouvertes.fr/hal-00472431

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, D. Mitsotakis, S. A. Beisel, and N. Y. Shokina, Dispersive waves generated by an underwater landslide, Numerical Methods for Hyperbolic Equations: Theory and Applications, pp.245-250, 2013.
DOI : 10.1201/b14172-33

URL : https://hal.archives-ouvertes.fr/hal-00637102

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

K. S. Erduran, S. Ilic, and V. Kutija, Hybrid finite-volume finite-difference scheme for the solution of Boussinesq equations, International Journal for Numerical Methods in Fluids, vol.128, issue.11, pp.1213-1232, 2005.
DOI : 10.1007/978-3-662-03490-3

J. Fenton, A ninth-order solution for the solitary wave, Journal of Fluid Mechanics, vol.26, issue.02, pp.257-271, 1972.
DOI : 10.1080/14786444908561410

E. D. Fernandez-nieto, F. Bouchut, D. Bresch, M. J. Castro-diaz, and A. Mangeney, A new Savage???Hutter type model for submarine avalanches and generated tsunami, Journal of Computational Physics, vol.227, issue.16, pp.7720-7754, 2008.
DOI : 10.1016/j.jcp.2008.04.039

URL : https://hal.archives-ouvertes.fr/hal-00385939

A. G. Filippini, S. Bellec, M. Colin, and M. Ricchiuto, On the nonlinear behaviour of Boussinesq type models: Amplitude-velocity vs amplitude-flux forms, Coastal Engineering, vol.99, issue.5, pp.109-123, 2015.
DOI : 10.1016/j.coastaleng.2015.02.003

URL : https://hal.archives-ouvertes.fr/hal-01140635

J. Ghidaglia, A. Kumbaro, and G. L. Coq, Une méthode volumes-finis à flux caractéristiques pour la résolution numérique des systèmes hyperboliques de lois de conservation, C. R. Acad. Sci. I, vol.322, pp.981-988, 1996.

A. E. Green and P. M. Naghdi, A derivation of equations for wave propagation in water of variable depth, Journal of Fluid Mechanics, vol.338, issue.02, pp.237-246, 1976.
DOI : 10.1017/S0022112076002425

S. T. Grilli and P. Watts, Modeling of waves generated by a moving submerged body. Applications to underwater landslides. Engineering Analysis with boundary elements, pp.645-656, 1999.

C. B. Harbitz, F. Lovholt, G. Pedersen, S. Glimsdal, and D. G. Masson, Mechanisms of tsunami generation by submarine landslides -a short review, Norwegian Journal of Geology, vol.86, issue.3, pp.255-264, 2006.

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

URL : http://hdl.handle.net/2060/19870018924

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/cgi-bin/GetTRDoc?AD=ADA158177&Location=U2&doc=GetTRDoc.pdf

N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM Philadelphia, p.14, 2002.
DOI : 10.1137/1.9780898718027

H. Kalisch, Stability of solitary waves for a nonlinearly dispersive equation. Discrete and Continuous Dynamical Systems, pp.709-717, 2004.

G. S. Khakimzyanov, D. Dutykh, and Z. I. Fedotova, Dispersive shallow water wave modelling . Part III: Model derivation on a globally spherical geometry, Commun. Comput. Phys, vol.23, issue.2 4, pp.315-360, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01547833

G. S. Khakimzyanov, D. Dutykh, Z. I. Fedotova, and D. E. Mitsotakis, Dispersive shallow water wave modelling. Part I: Model derivation on a globally flat space, Commun. Comput. Phys, vol.23, issue.1 4, pp.1-29, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01547833

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.

P. L. Liu, P. Lynett, and C. E. Synolakis, Analytical solutions for forced long waves on a sloping beach, Journal of Fluid Mechanics, vol.478, pp.101-109, 2003.
DOI : 10.1017/S0022112002003385

M. S. Longuet-higgins and J. Fenton, On the Mass, Momentum, Energy and Circulation of a Solitary Wave. II, Proc. R. Soc. A, pp.340471-493, 1623.
DOI : 10.1098/rspa.1974.0166

J. W. Lord-rayleigh, Waves, Phil. Mag, vol.1, pp.257-279
DOI : 10.1007/978-3-540-45626-1_8

P. A. Madsen, H. B. Bingham, and H. A. Schaffer, Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis, Proc. R. Soc. Lond. A, pp.1075-1104, 2003.
DOI : 10.1098/rspa.2002.1067

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

P. A. Madsen, H. A. Sorensen, and H. A. Schaffer, Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves, Coastal Engineering, vol.32, issue.4, pp.255-287, 1997.
DOI : 10.1016/S0378-3839(97)00028-8

O. Nwogu, Alternative Form of Boussinesq Equations for Nearshore Wave Propagation, Journal of Waterway, Port, Coastal, and Ocean Engineering, vol.119, issue.6, pp.618-638, 1993.
DOI : 10.1061/(ASCE)0733-950X(1993)119:6(618)

E. A. Okal and C. E. Synolakis, A Theoretical Comparison of Tsunamis from Dislocations and Landslides, Pure and Applied Geophysics, vol.160, issue.10-11, pp.2177-2188, 2003.
DOI : 10.1007/s00024-003-2425-x

E. A. Okal and C. E. Synolakis, Source discriminants for near-field tsunamis, Geophysical Journal International, vol.22, issue.45, pp.899-912, 2004.
DOI : 10.4294/jpe1952.22.415

URL : https://academic.oup.com/gji/article-pdf/158/3/899/5987920/158-3-899.pdf

P. J. Olver, Applications of Lie groups to differential equations, 1993.

F. Pascal, Sur des méthodes d'approximation effectives et d'analyse numérique pour les équations de la mécanique de fluides. Habilitation à diriger des recherches, p.23, 2002.

E. Pelinovsky and A. Poplavsky, Simplified model of tsunami generation by submarine landslides, Physics and Chemistry of the Earth, vol.21, issue.1-2, pp.13-17, 1996.
DOI : 10.1016/S0079-1946(97)00003-7

D. H. Peregrine, Long waves on a beach, Journal of Fluid Mechanics, vol.13, issue.04, pp.815-827, 1967.
DOI : 10.1029/JZ071i002p00393

J. Sandee and K. Hutter, On the development of the theory of the solitary wave. A historical essay, Acta Mechanica, vol.49, issue.3, pp.111-152, 1991.
DOI : 10.1080/14786447608639037

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

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

C. E. Synolakis, The runup of solitary waves, Journal of Fluid Mechanics, vol.87, issue.-1, pp.523-545, 1987.
DOI : 10.1017/S0022112058000331

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 : 10.1029/TR036i005p00759

S. Tinti, E. Bortolucci, and C. Chiavettieri, Tsunami Excitation by Submarine Slides in Shallow-water Approximation. Pure appl. geophys, pp.759-797, 2001.

V. V. Titov, F. I. Gonzalez, E. N. Bernard, M. C. Eble, H. O. Mofjeld et al., Real-Time Tsunami Forecasting: Challenges and Solutions, Natural Hazards, vol.35, issue.10, pp.41-58, 2005.
DOI : 10.1007/1-4020-3607-8_3

URL : http://nctr.pmel.noaa.gov/Dart/Pdf/titov_small.pdf

M. I. Todorovska, A. Hayir, and M. D. Trifunac, A note on tsunami amplitudes above submarine slides and slumps, Soil Dynamics and Earthquake Engineering, vol.22, issue.2, pp.129-141, 2002.
DOI : 10.1016/S0267-7261(01)00058-6

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 2009.
DOI : 10.1007/978-3-662-03490-3

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. Watts, F. Imamura, and S. T. Grilli, Comparing model simulations of three benchmark tsunami generation cases, Science of Tsunami Hazards, vol.18, issue.28, pp.107-123, 2000.

T. Y. Wu, Long Waves in Ocean and Coastal Waters, Journal of Engineering Mechanics, vol.107, pp.501-522, 1981.

T. Y. Wu, Generation of upstream advancing solitons by moving disturbances, Journal of Fluid Mechanics, vol.162, issue.-1, pp.75-99, 1987.
DOI : 10.1146/annurev.fl.12.010180.000303

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

J. Yang, Nonlinear Waves in Integrable and Nonintegrable Systems, Society for Industrial and Applied Mathematics, 2010.
DOI : 10.1137/1.9780898719680

J. A. Zelt, The run-up of nonbreaking and breaking solitary waves, Coastal Engineering, vol.15, issue.3, pp.205-246, 1991.
DOI : 10.1016/0378-3839(91)90003-Y

J. G. Zhou, D. M. Causon, D. M. Ingram, and C. G. Mingham, Numerical solutions of the shallow water equations with discontinuous bed topography, International Journal for Numerical Methods in Fluids, vol.33, issue.8, pp.769-788, 2002.
DOI : 10.1080/00221689509498555

A. Durán, Valladolid, Spain E-mail address: angel@mac.uva.es URL: https://www.researchgate.net/profile/Angel_Duran3, pp.47011-73000

F. Chambéry and L. , Campus Scientifique, F-73376 Le Bourget-du-Lac Cedex, France E-mail address: Denys.Dutykh@univ-smb.fr URL: http://www.denys-dutykh, Statistics and Operations Research