G. B. Alalykin, S. K. Godunov, L. L. Kireyeva, and L. A. Pliner, Solution of One-Dimensional Problems in Gas Dynamics on Moving Grids, 1970.

C. Arvanitis and A. I. Delis, Behavior of Finite Volume Schemes for Hyperbolic Conservation Laws on Adaptive Redistributed Spatial Grids, SIAM J. Sci. Comput, vol.28, issue.6, pp.1927-1956, 2006.

C. Arvanitis, T. Katsaounis, and C. Makridakis, Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws, ESAIM: Mathematical Modelling and Numerical Analysis, vol.35, issue.5, pp.17-33, 2010.
DOI : 10.1051/m2an:2001105

URL : https://www.esaim-m2an.org/articles/m2an/pdf/2001/01/m2an0033.pdf

R. Asselin, Frequency Filter for Time Integrations, Monthly Weather Review, vol.100, 1972.

B. N. Azarenok, S. A. Ivanenko, and T. Tang, Adaptive Mesh Redistibution Method Based on Godunov's Scheme, Commun. Math. Sci, vol.1, p.36, 2003.
DOI : 10.4310/cms.2003.v1.n1.a10

URL : http://www.intlpress.com/site/pub/files/_fulltext/journals/cms/2003/0001/0001/CMS-2003-0001-0001-a010.pdf

I. Babuska, W. D. Henshaw, J. E. Oliger, J. E. Flaherty, J. E. Hopcroft et al., Modeling, Mesh Generation, and Adaptive Numerical Methods for Partial Differential Equations, vol.75, 1995.

I. Babuska and M. Suri, The p and h-p Versions of the Finite Element Method, Basic Principles and Properties, SIAM Review, vol.36, issue.4, pp.578-632, 1994.

V. B. Barakhnin and N. V. Borodkin, The second order approximation TVD scheme on moving adaptive grids for hyperbolic systems, Sib. Zh. Vychisl. Mat, vol.3, 2000.
DOI : 10.1515/rnam-2001-0102

G. Beckett, J. A. Mackenzie, A. Ramage, and D. M. Sloan, Computational Solution of Two-Dimensional Unsteady PDEs Using Moving Mesh Methods, J. Comp. Phys, vol.182, issue.2, p.12, 2002.

M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, J. Comp. Phys, vol.82, issue.5, pp.64-84, 1989.
DOI : 10.1016/0021-9991(89)90035-1

URL : https://zenodo.org/record/1253914/files/article.pdf

M. J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Comp. Phys, vol.53, issue.3, pp.484-512, 1984.
DOI : 10.1016/0021-9991(84)90073-1

J. L. Bona and V. Varlamov, Wave generation by a moving boundary. Nonlinear partial differential equations and related analysis, vol.371, pp.41-71, 2005.
DOI : 10.1090/conm/371/06847

M. Breuss, About the Lax-Friedrichs scheme for the numerical approximation of hyperbolic conservation laws, PAMM, vol.4, issue.8, pp.636-637, 2004.

C. J. Budd, W. Huang, and R. D. Russell, Adaptivity with moving grids, Acta Numerica, vol.18, p.15, 2009.
DOI : 10.1017/s0962492906400015

C. J. Budd and J. F. Williams, Moving Mesh Generation Using the Parabolic MongeAmpère Equation, SIAM J. Sci. Comput, vol.31, pp.3438-3465, 2009.
DOI : 10.1137/080716773

J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, vol.1, pp.171-199, 1948.

W. Cao, W. Huang, and R. D. Russell, A Study of Monitor Functions for TwoDimensional Adaptive Mesh Generation, SIAM J. Sci. Comput, vol.20, issue.6, p.12, 1999.

M. Chhay, E. Hoarau, A. Hamdouni, and P. Sagaut, Comparison of some Liesymmetry-based integrators, J. Comp. Phys, vol.230, issue.5, pp.2174-2188, 2011.

R. Courant, K. Friedrichs, and H. Lewy, Über die partiellen Differenzengleichungen der mathematischen Physik, Mathematische Annalen, vol.100, p.25, 1928.

R. Courant, E. Isaacson, R. , and M. , On the solution of nonlinear hyperbolic differential equations by finite differences, Comm. Pure Appl. Math, vol.5, pp.243-255, 1952.

J. M. Coyle, J. E. Flaherty, L. , and R. , On the stability of mesh equidistribution strategies for time-dependent partial differential equations, J. Comp. Phys, vol.62, pp.26-39, 1986.

R. A. Dalrymple, S. T. Grilli, and J. T. Kirby, Tsunamis and challenges for accurate modeling, Oceanography, vol.19, issue.4, pp.142-151, 2006.

M. H. Dao and P. Tkalich, Tsunami propagation modelling -a sensitivity study, Nat. Hazards Earth Syst. Sci, vol.7, issue.4, pp.741-754, 2007.
URL : https://hal.archives-ouvertes.fr/hal-00299474

N. A. Dar'in, V. I. Mazhukin, and A. A. Samarskii, A finite-difference method for solving the equations of gas dynamics using adaptive nets which are dynamically associated with the solution, USSR Computational Mathematics and Mathematical Physics, vol.28, issue.4, pp.164-174, 1988.

P. Daripa, Iterative schemes and algorithms for adaptive grid generation in one dimension, J. Comp. Phys, vol.100, issue.2, pp.284-293, 1992.

L. M. Darmaev and V. D. Liseikin, A method of construction of multidimensional adaptive grids, Modelling in Mechanics, vol.1, issue.6, pp.49-58, 1987.

L. M. Degtyarev, V. V. Drozdov, and T. S. Ivanova, The method of adaptive grids for the solution of singularly perturbed one dimensional boundary value problems, Differ. Uravn, vol.23, issue.6, pp.1160-1169, 1987.

L. M. Degtyarev and T. S. Ivanova, The adaptive-grid method in one-dimensional nonstationary convection-diffusion problems, Differ. Uravn, vol.29, p.14, 1993.

E. A. Dorfi and L. O. Drury, Simple adaptive grids for 1 -D initial value problems, J. Comp. Phys, vol.69, pp.175-195, 1987.

V. A. Dorodnitsyn, Finite Difference Models Entirely Inheriting Continuous Symmetry of Original Differential Equations, Int. J. Mod. Phys. C, vol.05, p.4, 1994.

D. Dutykh, D. Clamond, P. Milewski, and D. Mitsotakis, Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations, Eur. J. Appl. Math, vol.24, p.36, 2013.
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 SystemsSeries B, vol.13, issue.4, pp.799-818, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00369795

D. Dutykh and E. Pelinovsky, Numerical simulation of a solitonic gas in KdV and KdV-BBM equations, Phys. Lett. A, vol.378, pp.3102-3110, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00913960

R. P. Fedorenko, Introduction to Computational Physics, 1994.

J. Fuhrmann, M. Ohlberger, and C. Rohde, Finite Volumes for Complex Applications VII -Elliptic, 2014.

S. Gasparin, J. Berger, D. Dutykh, and N. Mendes, An innovative method to determine optimum insulation thickness based on non-uniform adaptive moving grid, Journal of the Brazilian Society of Mechanical Sciences and Engineering, vol.41, issue.4, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02045015

D. L. George and R. J. Leveque, Finite volume methods and adaptive refinement for global tsunami propagation and local inundation, Sci. Tsunami Hazards, vol.24, issue.5, p.319, 2006.

S. Glimsdal, G. K. Pedersen, C. B. Harbitz, and F. Løvholt, Dispersion of tsunamis: does it really matter?, Natural Hazards and Earth System Science, vol.13, issue.6, pp.1507-1526, 2013.

S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb, vol.47, issue.5, pp.271-290, 1959.

S. K. Godunov, Reminiscences about Difference Schemes, J. Comput. Phys, vol.153, issue.5, pp.6-25, 1999.

S. K. Godunov, A. Zabrodin, M. Y. Ivanov, A. N. Kraiko, and G. P. Prokopov, Numerical solution of multidimensional problems of gas dynamics, 1976.

L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws: ExponentialFit, Well-Balanced and Asymptotic-Preserving, SIMAI Springer Series, vol.2, issue.26, 2013.

S. T. Grilli, M. Ioualalen, J. Asavanant, F. Shi, J. T. Kirby et al., Source Constraints and Model Simulation of the, Indian Ocean Tsunami. Journal of Waterway, Port, Coastal, and Ocean Engineering, vol.133, issue.4, pp.414-428, 2004.
URL : https://hal.archives-ouvertes.fr/hal-00407728

E. Hairer, S. P. Nørsett, and G. Wanner, Solving ordinary differential equations: Nonstiff problems, p.36, 2009.

A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comp. Phys, vol.49, pp.357-393, 1983.

P. Houston and E. Süli, hp-Adaptive Discontinuous Galerkin Finite Element Methods for First-Order Hyperbolic Problems, SIAM J. Sci. Comput, vol.23, issue.4, pp.1226-1252, 2001.

W. Huang, Y. Ren, and R. D. Russell, Moving Mesh Methods Based on Moving Mesh Partial Differential Equations, J. Comp. Phys, vol.113, issue.2, p.14, 1994.

W. Huang and R. D. Russell, Adaptive mesh movement -the MMPDE approach and its applications, J. Comp. Appl. Math, vol.128, issue.2, pp.383-398, 2001.

W. Huang and R. D. Russell, Applied Mathematical Sciences, vol.174, p.35, 2011.

F. Imamura, Simulation of wave-packet propagation along sloping beach by TUNAMI-code, pp.231-241, 1996.

G. Khakimzyanov and D. Dutykh, On supraconvergence phenomenon for second order centered finite differences on non-uniform grids, J. Comp. Appl. Math, vol.326, p.35, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01223522

G. S. Khakimzyanov and D. Dutykh, Numerical Modelling of Surface Water Wave Interaction with a Moving Wall, Commun. Comput. Phys, vol.23, p.36, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01546833

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.4, pp.315-360, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01547833

G. S. Khakimzyanov, D. Dutykh, and O. Gusev, Dispersive shallow water wave modelling. Part IV: Numerical simulation on a globally spherical geometry, Commun. Comput. Phys, vol.23, p.5, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01547833

G. S. Khakimzyanov, D. Dutykh, O. Gusev, and N. Y. Shokina, Dispersive shallow water wave modelling. Part II: Numerical modelling on a globally flat space, Commun. Comput. Phys, vol.23, p.36, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01547833

G. S. Khakimzyanov, N. Y. Shokina, D. Dutykh, and D. Mitsotakis, A new run-up algorithm based on local high-order analytic expansions, J. Comp. Appl. Math, vol.298, p.37, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01084811

J. T. Kirby, F. Shi, B. Tehranirad, J. C. Harris, and S. T. Grilli, Dispersive tsunami waves in the ocean: Model equations and sensitivity to dispersion and Coriolis effects, Ocean Modelling, vol.62, issue.4, pp.39-55, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01744889

E. Knobloch and R. Krechetnikov, Problems on Time-Varying Domains: Formulation, Dynamics, and Challenges, Acta Appl. Math, vol.137, issue.1, pp.123-157, 2015.

N. Krejic, T. Krunic, and M. Nedeljkov, Numerical verification of delta shock waves for pressureless gas dynamics, J. Math. Anal. Appl, vol.345, pp.243-257, 2008.

P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math, vol.13, issue.2, pp.217-237, 1960.

R. J. Leveque, Numerical Methods for Conservation Laws, 2 ed. Birkhäuser Basel, vol.9, p.27, 1992.

V. D. Liseikin, The construction of structured adaptive grids -a review, Comp. Math. Math. Phys, vol.36, issue.6, pp.1-32, 1996.

V. D. Liseikin, Grid Generation Methods, 2 ed. Scientific Computation, 2010.

F. Løvholt, G. Pedersen, and G. Gisler, Oceanic propagation of a potential tsunami from the La Palma Island, J. Geophys. Res, vol.113, p.9, 2008.

F. Lovholt, G. Pedersen, and S. Glimsdal, Coupling of Dispersive Tsunami Propagation and Shallow Water Coastal Response, The Open Oceanography Journal, vol.4, issue.1, pp.71-82, 2010.

R. W. Maccormack, The effect of viscosity in hypervelocity impact cratering, pp.69-354, 1969.

P. G. Michalopoulos, P. Yi, and A. S. Lyrintzis, Continuum modelling of traffic dynamics for congested freeways, Transportation Research Part B: Methodological, vol.27, issue.5, pp.315-332, 1993.

N. R. Mirchina and E. N. Pelinovsky, Nonlinear and dispersive effects for tsunami waves in the open ocean, Int. J. Tsunami Soc, vol.2, issue.4, pp.1073-1081, 1982.

T. S. Murty, A. D. Rao, N. Nirupama, and I. Nistor, Numerical modelling concepts for tsunami warning systems, Current Science, vol.90, issue.4, pp.1073-1081, 2006.

D. H. Peregrine, Long waves on a beach, J. Fluid Mech, vol.27, issue.4, pp.815-827, 1967.

S. Popinet, Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries, J. Comp. Phys, vol.190, 2003.
URL : https://hal.archives-ouvertes.fr/hal-01445436

G. P. Prokopov, About organization of comparison of algorithms and programs for 2D regular difference mesh construction, Vopr. At. Nauki Tekh., Ser. Mat. Model. Fiz. Processov, vol.3, issue.6, pp.98-108, 1989.

P. L. Roe, Approximate Riemann solvers, parameter vectors and difference schemes, J. Comput. Phys, vol.43, p.24, 1981.

B. L. Rozhdestvenskiy and N. N. Yanenko, Systems of quasilinear equations and their application to gas dynamics, Nauka, vol.10, p.28, 1978.

T. I. Serezhnikova, A. F. Sidorov, and O. V. Ushakova, On one method of construction of optimal curvilinear grids and its applications, Russ. J. Numer. Anal. Math. Modelling, vol.4, issue.6, pp.137-156, 1989.

Y. I. Shokin, V. V. Babailov, S. A. Beisel, L. B. Chubarov, S. V. Eletsky et al., Mathematical Modeling in Application to Regional Tsunami Warning Systems Operations, Computational Science and High Performance Computing III, pp.52-68, 2008.

Y. I. Shokin, Y. V. Sergeeva, and G. S. Khakimzyanov, Construction of monotonic schemes by the differential approximation method, Russ. J. Numer. Anal. Math. Modelling, vol.20, p.11, 2005.

Y. I. Shokin and A. I. Urusov, On the construction of adaptive algorithms for unsteady problems of gas dynamics in arbitrary coordinate systems, Eighth International Conference on Numerical Methods in Fluid Dynamics, vol.6, p.14, 1982.

Y. I. Shokin and N. N. Yanenko, Method of Differential Approximation. Application to Gas Dynamics, 1985.

J. M. Stockie, J. A. Mackenzie, and R. D. Russell, A Moving Mesh Method for One-dimensional Hyperbolic Conservation Laws, SIAM J. Sci. Comput, vol.22, p.21, 2001.

J. J. Stoker, Water Waves: The Mathematical Theory with Applications, 1920.

V. G. Sudobicher and S. M. Shugrin, Flow along a dry channel, Izv. Akad. Nauk SSSR, vol.13, issue.6, pp.116-122, 1968.

C. E. Synolakis and E. N. Bernard, Tsunami science before and beyond Boxing Day, Phil. Trans. R. Soc. A, vol.364, issue.4, pp.2231-2265, 2004.

H. Tang and T. Tang, Adaptive Mesh Methods for One-and Two-Dimensional Hyperbolic Conservation Laws, SIAM J. Numer. Anal, vol.41, issue.2, p.14, 2003.

P. D. Thomas and C. K. Lombart, Geometric conservation law and its application to flow computations on moving grid, AIAA Journal, vol.17, p.26, 1979.

V. V. Titov and F. I. González, Implementation and testing of the method of splitting tsunami (MOST) model, 1997.

V. V. Titov and C. E. Synolakis, Numerical modeling of 3-D long wave runup using VTCS-3, Long wave runup models, pp.242-248, 1996.

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, p.28, 2009.

A. Van-dam and P. A. Zegeling, A robust moving mesh finite volume method applied to 1D hyperbolic conservation laws from magnetohydrodynamics, J. Comp. Phys, vol.216, issue.2, pp.526-546, 2006.

N. E. Voltsinger, E. N. Pelinovsky, and K. A. Klevannyi, Long wave dynamics of coastal regions, Gidrometeoizdat, p.28, 1989.

X. Wang and W. L. Power, COMCOT: a tsunami generation, propagation and run-up model, GNS Science, 2011.

G. B. Whitham, Linear and nonlinear waves, vol.5, p.28, 1999.

P. A. Zegeling and J. G. Blom, An evaluation of the gradient-weighted movingfinite-element method in one space dimension, J. Comp. Phys, vol.103, issue.2, pp.422-441, 1992.

P. A. Zegeling, J. G. Verwer, and J. C. Van-eijkeren, Application of a moving grid method to a class of 1D brine transport problems in porous media, Int. J. Num. Meth. Fluids, vol.15, pp.175-191, 1992.

G. Khakimzyanov, Institute of Computational Technologies of Siberian Branch of the Russian Academy of Sciences, Novosibirsk 630090, Russia E-mail address, Khak@ict.nsc.ru URL

D. Dutykh, , p.73000

F. Chambéry and L. , , p.73376

L. Bourget-du-lac, . Cedex, E. France, and . Address, Denys.Dutykh@univ-smb.fr URL, vol.600

N. Yu and . Shokina, Nina.Shokina@gmail.com URL