EDP Sciences logo
Subscriber Authentication Point
Search
Menu
All issues Volume 58 / No 4 (July-August 2024) ESAIM: M2AN, 58 4 (2024) 1497-1522 References
Open Access
Issue
ESAIM: M2AN
Volume 58, Number 4, July-August 2024
Page(s) 1497 - 1522
DOI https://doi.org/10.1051/m2an/2024048
Published online 27 August 2024
  1. N. Aissiouene, M.-O. Bristeau, E. Godlewski and J. Sainte-Marie, A combined finite volume – finite element scheme for a dispersive shallow water system. Netw. Heterogen. Media (NHM) 11 (2016) 1–27. [CrossRef] [Google Scholar]
  2. N. Andrianov, Performance of numerical methods on the non-unique solution to the Riemann problem for the shallow water equations. Int. J. Numer. Methods Fluids 47 (2005) 825–831. [CrossRef] [Google Scholar]
  3. E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050–2065. [Google Scholar]
  4. T.B. Benjamin, J.L. Bona and J.J. Mahony, Model equations for long waves in nonlinear dispersive systems. Philos. Trans. R. Soc. London Ser. A Math. Phys. Sci. 272 (1997) 47–78. [Google Scholar]
  5. C. Besse, S. Gavrilyuk, M. Kazakova and P. Noble, Perfectly matched layers methods for mixed hyperbolic–dispersive equations. Water Waves 4 (2022) 313–343. [Google Scholar]
  6. 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]
  7. F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws, and Well-Balanced Schemes for Sources. Springer Science & Business Media (2004). [CrossRef] [Google Scholar]
  8. B. Boutin, F. Coquel and P.G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. I. Self-similar diffusion for thin interfaces. Proc. R. Soc. Edinburgh Sect. A: Math. 141 (2011) 921–956. [CrossRef] [Google Scholar]
  9. B. Boutin, F. Coquel and P.G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces. SIAM J. Numer. Anal. 51 (2013) 1108–1133. [CrossRef] [MathSciNet] [Google Scholar]
  10. B. Boutin, F. Coquel and P.G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. IV. Well-balanced schemes for scalar multi-dimensional and multi-component laws. Math. Comput. 84 (2015) 1663–1702. [CrossRef] [Google Scholar]
  11. B. Boutin, F. Coquel and P.G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. II. Resonant interfaces with internal structure. Netw. Heterogen. Media 16 (2021) 283. [CrossRef] [Google Scholar]
  12. D. Bresch, D. Lannes and G. Métivier, Waves interacting with a partially immersed obstacle in the Boussinesq regime. Anal. PDE 14 (2021) 1085–1124. [CrossRef] [MathSciNet] [Google Scholar]
  13. R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993) 1661–1664. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  14. R. Cienfuegos, E. Barthélemy and P. Bonneton, A fourth-order compact finite volume scheme for fully nonlinear and weakly dispersive Boussinesq-type equations. Part I: model development and analysis. Int. J. Numer. Methods Fluids 51 (2006) 1217–1253. [CrossRef] [Google Scholar]
  15. V. Duchêne, Many Models for Water Waves. Habilitation à diriger des recherches, Université de Rennes 1 (2021). [Google Scholar]
  16. 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]
  17. 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. [CrossRef] [MathSciNet] [Google Scholar]
  18. 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]
  19. J.-F. Gerbeau and B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water: numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 89–102. [MathSciNet] [Google Scholar]
  20. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Vol. 118 of Applied Mathematical Sciences. Springer-Verlag, New York (1996). [CrossRef] [Google Scholar]
  21. E. Godlewski, K.-C.L. Thanh and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws : II. The case of systems. ESAIM: Math. Modell. Numer. Anal. 39 (2005) 649–692. [CrossRef] [EDP Sciences] [Google Scholar]
  22. E. Godlewski, M. Parisot, J. Sainte-Marie and F. Wahl, Congested shallow water model: roof modeling in free surface flow. ESAIM: Math. Modell. Numer. Anal. 52 (2018) 1679–1707. [CrossRef] [EDP Sciences] [Google Scholar]
  23. M.F. Göz and C.-D. Munz, Approximate Riemann Solvers for Fluid Flow with Material Interfaces. Springer Netherlands, Dordrecht (1998) 211–235. [Google Scholar]
  24. A.E. Green and P.M. Naghdi, A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78 (1976) 237–246. [Google Scholar]
  25. J. Guermond, P. Minev and J. Shen, An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195 (2006) 6011–6045. [Google Scholar]
  26. M. Kazakova and P. Noble, Discrete transparent boundary conditions for the linearized Green–Naghdi system of equations. SIAM J. Numer. Anal. 58 (2020) 657–683. [CrossRef] [MathSciNet] [Google Scholar]
  27. M. Kazolea and M. Ricchiuto, Full nonlinearity in weakly dispersive Boussinesq models: luxury or necessity. J. Hydraulic Eng. 150 (2024) 04023061. [CrossRef] [Google Scholar]
  28. M. Kazolea, A. Delis and C. Synolakis, Numerical treatment of wave breaking on unstructured finite volume approximations for extended Boussinesq-type equations. J. Comput. Phys. 271 (2014) 281–305. [CrossRef] [MathSciNet] [Google Scholar]
  29. D.J. Korteweg and G. de Vries, XlI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. London Edinburgh Dublin Philos. Mag. J. Sci. 39 (1895) 422–443. [CrossRef] [Google Scholar]
  30. D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics. Vol. 188. Mathematical Surveys and Monographs. American Mathematical Society (2013). [CrossRef] [Google Scholar]
  31. D. Lannes and B. Alvarez-Samaniego, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green–Naghdi equations. Indiana Univ. Math. J. 57 (2008) 97–132. [CrossRef] [MathSciNet] [Google Scholar]
  32. 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]
  33. D. Lannes and M. Rigal, General boundary conditions for a Boussinesq model with varying bathymetry. Preprint 2402.03859 (2024). [Google Scholar]
  34. D. Lannes and L. Weynans, Generating boundary conditions for a Boussinesq system. Nonlinearity 33 (2020) 6868–6889. [CrossRef] [MathSciNet] [Google Scholar]
  35. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Vol. 31. Cambridge University Press (2002). [CrossRef] [Google Scholar]
  36. M. Li, P. Guyenne, F. Li and L. Xu, High order well-balanced CDG–FE methods for shallow water waves by a Green–Naghdi model. J. Comput. Phys. 257 (2014) 169–192. [CrossRef] [MathSciNet] [Google Scholar]
  37. P.A. Madsen and H.A. Sch¨affer, Higher–order Boussinesq–type equations for surface gravity waves: derivation and analysis. Philos. Trans. R. Soc. London Ser. A: Math. Phys. Eng. Sci. 356 (1998) 3123–3181. [CrossRef] [MathSciNet] [Google Scholar]
  38. P.A. Madsen, R. Murray and O.R. Sørensen, A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal Eng. 15 (1991) 371–388. [CrossRef] [Google Scholar]
  39. Y. Matsuno, Hamiltonian formulation of the extended Green–Naghdi equations. Phys. D: Nonlinear Phenom. 301, 302 (2015) 1–7. [CrossRef] [Google Scholar]
  40. 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]
  41. 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]
  42. M. Parisot, Entropy-satisfying scheme for a hierarchy of dispersive reduced models of free surface flow. Int. J. Numer. Methods Fluids 91 (2019) 509–531. [CrossRef] [Google Scholar]
  43. D.H. Peregrine, Long waves on a beach. J. Fluid Mech. 27 (1967) 815–827. [CrossRef] [Google Scholar]
  44. M. Ricchiuto and A. Filippini, Upwind residual discretization of enhanced Boussinesq equations for wave propagation over complex bathymetries. J. Comput. Phys. 271 (2014) 306–341. [CrossRef] [MathSciNet] [Google Scholar]
  45. G.L. Richard, A. Duran and B. Fabrèges, A new model of shoaling and breaking waves. Part II. Run-up and two-dimensional waves. J. Fluid Mech. 867 (2019) 146–194. [CrossRef] [MathSciNet] [Google Scholar]
  46. E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer Science & Business Media (2013). [Google Scholar]
  47. G. Wei, J.T. Kirby and A. Sinha, Generation of waves in Boussinesq models using a source function method. Coastal Eng. 36 (1999) 271–299. [CrossRef] [Google Scholar]
  48. Y. Yamazaki, Z. Kowalik and K.F. Cheung, Depth-integrated, non-hydrostatic model for wave breaking and run-up. Int. J. Numer. Methods Fluids 61 (2009) 473–497. [CrossRef] [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