EDP Sciences logo
Subscriber Authentication Point
Sciengine logo
Search
Menu
All issues Volume 60 / No 3 (May-June 2026) ESAIM: M2AN, 60 3 (2026) 1411-1449 References
Open Access
Issue
ESAIM: M2AN
Volume 60, Number 3, May-June 2026
Page(s) 1411 - 1449
DOI https://doi.org/10.1051/m2an/2026026
Published online 05 June 2026
  1. M. Antuono, V. Liapidevskii and M. Brocchini, Dispersive nonlinear shallow-water equations. Stud. Appl. Math. 122 (2009) 1–28. [Google Scholar]
  2. S. Beji and J. Battjes, Experimental investigation of wave propagation over a bar. Coast. Eng. 80 (1993) 151–162. [Google Scholar]
  3. C. Berthon, A. Duran, F. Foucher, K. Saleh and J.D.D. Zabsonré, Improvement of the hydrostatic reconstruction scheme to get fully discrete entropy inequalities. J. Sci. Comput. 80 (2019) 924–956. [CrossRef] [MathSciNet] [Google Scholar]
  4. J.L. Bona, M. Chen and J.C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: derivation and linear theory. J. Nonlinear Sci. 12 (2002) 283–318. [CrossRef] [MathSciNet] [Google Scholar]
  5. P. Bonneton, F. Chazel, D. Lannes, F. Marche and M. Tissier, A splitting approach for the fully nonlinear and weakly dispersive Green-Naghdi model. J. Comput. Phys. 230 (2011) 1479–1498. [Google Scholar]
  6. J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. Journal de mathématiques pures et appliquées 17 (1872) 55–108. [Google Scholar]
  7. M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic shallow water model. Comput. Fluids 47 (2011) 51–64. [Google Scholar]
  8. S. Busto, M. Dumbser, C. Escalante, N. Favrie and S. Gavrilyuk, On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems. J. Sci. Comput. 87 (2021) 48. [Google Scholar]
  9. V. Casulli and E. Cattani, Stability, accuracy and efficiency of a semi-implicit method for three-dimensional shallow water flow. Comput. Math. App. 27 (1994) 99–112. [Google Scholar]
  10. V. Casulli and P. Zanolli, Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems. Math. Comput. Model. 36 (2002) 1131–1149. [Google Scholar]
  11. R. Chassagne, A.G. Filippini, M. Ricchiuto and P. Bonneton, Dispersive and dispersive-like bores in channels with sloping banks. J. Fluid Mech. 870 (2019) 595–616. [Google Scholar]
  12. H. Chanson, Current knowledge in hydraulic jumps and related phenomena. A survey of experimental results. Eur. J. Mech. B/Fluids 28 (2009) 191–210. [Google Scholar]
  13. F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model. J. Sci. Comput. 48 (2011) 105–116. [Google Scholar]
  14. A. Chesnokov and T.H. Nguyen, Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography. Comput. Fluids 189 (2019) 13–23. [Google Scholar]
  15. D. Clamond, D. Dutykh and D. Mitsotakis, Conservative modified Serre-Green-Naghdi equations with improved dispersion characteristics. Commun. Nonlinear Sci. Numer. Simul. 45 (2017) 245–257. [Google Scholar]
  16. F. Couderc, A. Duran and J.P. Vila, An explicit asymptotic preserving low Froude scheme for the multilayer shallow water model with density stratification. J. Comput. Phys. 343 (2017) 235–270. [CrossRef] [MathSciNet] [Google Scholar]
  17. J.W. Daily and S.C. Stephan, Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model. Coast. Eng. Proc. 1 (1952) 2. [Google Scholar]
  18. A. Del Grosso, M.J. Castro, A. Chan, G. Gallice, R. Loubère and P.-H. Maire, A well-balanced, positive, entropy-stable, and multi-dimensional-aware finite volume scheme for 2D shallow-water equations with unstructured grids. J. Comput. Phys. 503 (2024) 112829. [Google Scholar]
  19. J.S.A. Do Carmo, J.A. Ferreira, L. Pinto and G. Romanazzi, An improved Serre model: efficient simulation and comparative evaluation. Appl. Math. Model. 56 (2018) 404–423. [Google Scholar]
  20. H. Dong and M. Li, A reconstructed central discontinuous Galerkin-finite element method for the fully nonlinear weakly dispersive Green-Naghdi model. Appl. Numer. Math. 110 (2016) 110–127. [Google Scholar]
  21. V. Duchêne, Rigorous justification of the Favrie-Gavrilyuk approximation to the Serre-Green-Naghdi model. Nonlinearity 32 (2019) 3772. [Google Scholar]
  22. V. Duchêne, Many Models for Water Waves. HDR, Université de Rennes 1 (2021). [Google Scholar]
  23. A. Duran, Revisiting energy estimates of the CPR scheme for the Shallow Water equations. Submitted, hal preprint available at https://hal.science/hal-04220336 (2023). [Google Scholar]
  24. A. Duran and F. Marche, Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations. Commun. Comput. Phys. 17 (2015) 721–760. [CrossRef] [MathSciNet] [Google Scholar]
  25. A. Duran and F. Marche, Discontinuous Galerkin discretization of Green-Naghdi equations on unstructured simplicial meshes. Appl. Math. Model. 45 (2017) 840–864. [Google Scholar]
  26. A. Duran and G.L. Richard, Modelling coastal wave trains and wave breaking. Ocean Model. 147 (2020) 101581. [Google Scholar]
  27. L. Emerald, Rigorous derivation from the water waves equations of some full dispersion shallow water models. SIAM J. Math. Anal. 53 (2021) 3772–3800. [Google Scholar]
  28. C. Escalante and T. Morales de Luna, A general non-hydrostatic hyperbolic formulation for Boussinesq dispersive shallow flows and its numerical approximation. J. Sci. Phys. 83 (2020) 1–37. [Google Scholar]
  29. C. Escalante, M. Dumbser and M. Castro, An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous galerkin schemes. J. Comput. Phys. 394 (2019) 385–419. [Google Scholar]
  30. C. Escalante, E.D. Fernández-Nieto, J. Garres-Díaz, T. Morales de Luna and Y. Penel, Non-hydrostatic layer-averaged approximation of Euler system with enhanced dispersion properties. Comput. Appl. Math. 42 (2023) 177. [CrossRef] [Google Scholar]
  31. H. Favre, Etude théorique et expérimentale des ondes de translation dans les canaux découverts. Dunod, Paris (1935). [Google Scholar]
  32. N. Favrie and S. Gavrilyuk, A rapid numerical method for solving Serre-Green-Naghdi equations describing long free surface gravity waves. Nonlinearity 30 (2017) 2718–2736. [CrossRef] [MathSciNet] [Google Scholar]
  33. E.D. Fernández-Nieto, M. Parisot, Y. Penel and J. Sainte-Marie, A hierarchy of dispersive layer-averaged approximations of Euler equations for free surface flows. Commun. Math. Sci. 16 (2018) 1169–1202. [CrossRef] [MathSciNet] [Google Scholar]
  34. S. Gavrilyuk and K.-M. Shyue, 2D Serre Green-Naghdi equations over topography: elliptic operator inversion method. J. Fluid Mech. 150 (2023) 237–246. [Google Scholar]
  35. S.L. Gavrilyuk, V.Yu. Liapidevskii and A.A. Chesnokov, Spilling breakers in shallow water: applications to Favre waves and to the shoaling and breaking of solitary waves. J. Fluid Mech. 808 (2016) 441–468. [Google Scholar]
  36. A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78 (1976) 237–246. [CrossRef] [Google Scholar]
  37. G. Grosso, M. Antuono and M. Brocchini, Dispersive nonlinear shallow-water equations: some preliminary numerical results. J. Eng. Math. 67 (2010) 71–84. [Google Scholar]
  38. J.L. Guermond, B. Popov, E. Tovar and C. Kees, Robust explicit relaxation technique for solving the Green-Naghdi equations. J. Comput. Phys. 399 (2019) 108917. [Google Scholar]
  39. J.L. Guermond, C. Kees, B. Popov and E. Tovar, Hyperbolic relaxation technique for solving the dispersive Serre-Green-Naghdi equations with topography. J. Comput. Phys. 450 (2022) 110809. [Google Scholar]
  40. B. Jouy, D. Violeau, M. Ricchiuto and M. Le, One dimensional modelling of Favre waves in channels. Appl. Math. Model. 133 (2024) 170–194. [Google Scholar]
  41. M. Kazakova and G.L. Richard, A new model of shoaling and breaking waves: one-dimensional solitary wave on a mild sloping beach. J. Fluid Mech. 862 (2019) 552–591. [Google Scholar]
  42. M. Kazolea and M. Ricchiuto, On wave breaking for Boussinesq-type models. Ocean Modell. 123 (2018) 16–39. [Google Scholar]
  43. M. Kazolea, A.G. Filippini and M. Ricchiuto, Low dispersion finite volume/element discretization of the enhanced Green-Naghdi equations for wave propagation, breaking and runup on unstructured meshes. Ocean Modell. 182 (2023) 102157. [Google Scholar]
  44. S. Kivva, Entropy stable flux correction for hydrostatic reconstruction scheme for shallow water flows. J. Sci. Comput. 99 (2024) 1. [Google Scholar]
  45. D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics. Vol. 188. American Mathematical Society (2013). [Google Scholar]
  46. D. Lannes and F. Marche, A new class of fully nonlinear and weakly dispersive green-naghdi models for efficient 2d simulations. J. Comput. Phys. 282 (2015) 238–268. [CrossRef] [MathSciNet] [Google Scholar]
  47. O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model. J. Comput. Phys. 229 (2010) 2034–2045. [Google Scholar]
  48. Z. Liu and Z. Sun, Two sets of higher-order Boussinesq-type equations for water waves. Ocean Eng. 32 (2005) 1296–1310. [Google Scholar]
  49. Z. Liu, K. Fang and Y. Cheng, A new multi-layer irrotational boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed. J. Fluid Mech. 842 (2018) 323–353. [Google Scholar]
  50. M. Li, L. Xu and Y. Cheng, A CDG-FE method for the two-dimensional Green-Naghdi model with the enhanced dispersive property. J. Comput. Phys. 399 (2019) 108953. [Google Scholar]
  51. R. Lteif, An operator-splitting approach with a hybrid finite volume/finite difference scheme for extended Boussinesq models. Appl. Numer. Math. 196 (2024) 159–182. [Google Scholar]
  52. P.A. Madsen and O.R. Sørensen, A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly-varying bathymetry. Coast. Eng. 18 (1992) 183–204. [Google Scholar]
  53. P.A. Madsen, B. Banijamali, H.A. Schäffer and O.R. Sørensen, Boussinesq type equations with high accuracy in dispersion and nonlinearity, in Coastal Engineering (1996) 95–108. [Google Scholar]
  54. P.A. Madsen, H.B. Bingham and H. Liu, A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech. 462 (2002) 1–30. [Google Scholar]
  55. F. Marche, A combined Hybridized Discontinuous Galerkin (HDG) and Discontinuous Galerkin (DG) discrete formulation for Green-Naghdi equations on unstructured meshes. J. Comput. Phys. 418 (2020) 109637. [Google Scholar]
  56. D. Mitsotakis, B. Ilan and D. Dutykh, On the Galerkin/finite-element method for the serre equations. J. Sci. Comput. 61 (2014) 166–195. [CrossRef] [MathSciNet] [Google Scholar]
  57. S. Noelle, M. Parisot and T. Tscherpel, A class of boundary conditions for time-discrete Green-Naghdi equations with bathymetry. SIAM J. Numer. Anal. 60 (2022) 2681–2712. [CrossRef] [MathSciNet] [Google Scholar]
  58. O. Nwogu, Alternative form of boussinesq equations for nearshore wave propagation. J. Waterway Port Coast. Ocean Eng. 119 (1993) 618–638. [Google Scholar]
  59. N. Panda, C. Dawson, Y. Zhang, A.B. Kennedy, J.J. Westerink and A.S. Donahue, Discontinuous Galerkin methods for solving Boussinesq-Green-Naghdi equations in resolving non-linear and dispersive surface water waves. J. Comput. Phys. 273 (2014) 572–588. [Google Scholar]
  60. M. Parisot and J.P. Vila, Centered-potential regularization of advection upstream splitting method: application to the multilayer shallow water model in the low Froude number regime. SIAM J. Numer. Anal. 54 (2016) 3083–3104. [Google Scholar]
  61. D.H. Peregrine, Long waves on a beach. J. Fluid Mech. 27 (1967) 815–827. [CrossRef] [Google Scholar]
  62. J.P.A. Pitt, C. Zoppou and S.G. Roberts, Numerical scheme for the generalised Serre-Green-Naghdi model. Wave Motion 115 (2022) 103077. [Google Scholar]
  63. S. Popinet, A quadtree-adaptive multigrid solver for the Serre-Green-Naghdi equations. J. Comput. Phys. 302 (2015) 336–358. [Google Scholar]
  64. G.L. Richard, An extension of the Boussinesq-type models to weakly compressible flows. Eur. J. Mech. B/Fluids 89 (2021) 217–240. [Google Scholar]
  65. G.L. Richard, M. Gisclon, C. Ruyer-Quil and J.-P. Vila, Optimization of consistent two-equation models for thin film flows. Eur. J. Mech. B/Fluids 76 (2019) 7–25. [Google Scholar]
  66. M. Rigal, C. El Hassanieh and J. Sainte-Marie, Implicit kinetic schemes for the Saint-Venant system. ESAIM: Math. Modell. Numer. Anal. 59 (2025) 1863–1908. [Google Scholar]
  67. A. Samii and C. Dawson, An explicit hybridized discontinuous Galerkin method for Serre-Green-Naghdi wave model. Comput. Methods Appl. Mech. Eng. 330 (2018) 447–470. [Google Scholar]
  68. F. Serre, Contribution à l'étude des écoulements permanents et variables dans les canaux. La Houille Blanche 6 (1953) 830–872. [Google Scholar]
  69. S. Soares-Frazão and Y. Zech, Undular bores and secondary waves experiments and hybrid finite-volume modeling. J. Hydraulic Res. 1 (2002) 33–43. [Google Scholar]
  70. M. Tissier, P. Bonneton, F. Marche, F. Chazel and D. Lannes, Nearshore dynamics of tsunami-like undular bores using a fully nonlinear Boussinesq model. J. Coast. Res. 64 (2011) 1478–1503. [Google Scholar]
  71. M. Tissier, P. Bonneton, F. Marche, F. Chazel and D. Lannes, A new approach to handle wave breaking in fully non-linear Boussinesq models. Coast. Eng. 67 (2012) 54–66. [Google Scholar]
  72. S. Tkachenko, S. Gavrilyuk and J. Massoni, Extended Lagrangian approach for the numerical study of multidimensional dispersive waves: applications to the Serre Green-Naghdi equations. J. Comput. Phys. 477 (2023) 111901. [Google Scholar]
  73. M. Tonelli and M. Petti, Hybrid finite volume finite difference scheme for 2DH improved Boussinesq equations. Coast. Eng. 56 (2009) 609–620. [Google Scholar]
  74. A. Treske, Undular bores (favre-waves) in open channels-experimental studies. J. Hydraulic Res. 32 (1994) 355–370. [Google Scholar]
  75. G. Wei, J.T. Kirby, S.T. Grilli and R. Subramanya, A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear unsteady waves. J. Fluid Mech. 294 (1995) 71–92. [Google Scholar]
  76. H. Yen Chung, The enhanced breaking wave model and the morphodynamic coupling. Ph.D. thesis, Université Savoie Mont-Blanc (2025). [Google Scholar]
  77. M. Zefzouf and F. Marche, A new symmetric interior penalty discontinuous Galerkin formulation for the Serre-Green-Naghdi equations. Numer. Methods Part. Differ. Eq. 39 (2023) 1478–1503. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you