A 1D Stabilized Finite Element Model for Non-hydrostatic Wave Breaking and Run-up

  • P. BacigaluppiEmail author
  • M. Ricchiuto
  • P. Bonneton
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 78)


We present a stabilized finite element model for wave propagation, breaking and run-up. Propagation is modelled by a form of the enhanced Boussinesq equations, while energy transformation in breaking regions is captured by reverting to the shallow water equations and allowing waves to locally converge into discontinuities. To discretize the system we propose a non-linear variant of the stabilized finite element method of (Ricchiuto and Filippini, J.Comput.Phys. 2014). To guarantee monotone shock capturing, a non-linear mass-lumping procedure is proposed which locally reverts the third order finite element scheme to the first order upwind scheme. We present different definitions of the breaking criterion, including a local implementation of the convective criterion of (Bjørkavåg and Kalisch, Phys.Letters A 2011), and discuss in some detail the implementation of the shock capturing technique. The robustness of the scheme and the behaviour of different breaking criteria is investigated on several cases with available experimental data.


Wave Height Wave Breaking Shallow Water Equation Boussinesq Equation Free Surface Velocity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Bacigaluppi, P.: Upwind stabilized finite element modeling of non-hydrostatic wave breaking and run-up (2013). MSc Thesis, Aerospace Department, Politecnico di MilanoGoogle Scholar
  2. 2.
    Barthélemy, E.: Nonlinear shallow water theories for coastal waves. Surv. Geophys. 25, 315–337 (2004)CrossRefGoogle Scholar
  3. 3.
    Bjørkavåg, M., Kalisch, H.: Wave breaking in boussinesq models for undular bores. Phys. Lett. A 375(14), 1570–1578 (2011)Google Scholar
  4. 4.
    Bonneton, P.: Modelling of periodic wave transformation in the inner surf zone. Ocean Eng. 34, 1459–1471 (2007)CrossRefGoogle Scholar
  5. 5.
    Bonneton, P., Chazel, F., Lannes, D., Marche, F., Tissier, M.: A splitting approach for the fully nonlinear and weakly dispersive greennaghdi model. J. Comput. Phys. 230(4) (2011)Google Scholar
  6. 6.
    Castro, M., Ferrero, A., García-Rodríguez, J., González-Vida, J., Macías, J., Pareés, C., Vázquez-Cendón, M.: The numerical treatment of wet/dry fronts in shallow flows: application to one-layer and two-layer systems. Math. Comput. Model. 42, 419–439 (2005)CrossRefzbMATHGoogle Scholar
  7. 7.
    Dingemans, M.: Water Wave Propagation Over Uneven Bottoms: Linear wave propagation. Advanced series on ocean engineering. World Scientific Pub, Singapore (1997)Google Scholar
  8. 8.
    Hansen, J., Svendsen, I.: Regular waves in shoaling water: experimental data. Tech. Rep. 21, Technical Report, ISVA series paper (1979)Google Scholar
  9. 9.
    Harten, A., Hyman, J.: Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50(2) (1983)Google Scholar
  10. 10.
    Kazolea, M., Delis, A., Synolakis, C.: Numerical treatment of wave breaking on unstructured finite volume approximations for extended boussinesq-type equations. J. Comput. Phys. (2014).
  11. 11.
    LeMétayer, O., Gavrilyuk, S., Hank, S.: A numerical scheme for the green-naghdi model. J. Comput. Phys. 229(6) (2010)Google Scholar
  12. 12.
    LeVeque, R.: Finite-Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2004)Google Scholar
  13. 13.
    Ma, G., Shi, F., Kirby, J.: Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Model 43–44, 22–35 (2012)CrossRefGoogle Scholar
  14. 14.
    Madsen, P., Sørensen, O.: A new form of the Boussinesq equations with improved linear dispersion characteristics. A slowly-varying bathymetry. Coast. Eng. 18, 183–204 (1992)Google Scholar
  15. 15.
    Ricchiuto, M., Bollermann, A.: Stabilized Residual Distribution for shallow water simulations. J. Comput. Phys. 228, 1071–1115 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Ricchiuto, M., Filippini, A.: Upwind residual discretization of enhanced boussinesq equations for wave propagation over complex bathymetries. J. Comput. Phys. (2014).
  17. 17.
    Shiach, J., Mingham, C.: A temporally second-order accurate Godunov-type scheme for solving the extended Boussinesq equations. Coast. Eng. 56, 32–45 (2009)CrossRefGoogle Scholar
  18. 18.
    Tissier, M., Bonneton, P., Marche, F., Chazel, F., Lannes, D.: A new approach to handle wave breaking in fully non-linear boussinesq models. Coast. Eng. 67, 54–66 (2012)CrossRefGoogle Scholar
  19. 19.
    Tonelli, M., Petti, M.: Simulation of wave breaking over complex bathymetries by a Boussinesq model. J. Hydraulic Res. 49 (2011)Google Scholar
  20. 20.
    Veeramony, J., Svendsen, I.: The flow in surf-zone waves. Coast. Eng. 39, 93–122 (2000)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Inria Bordeaux Sud-OuestTalence CedexFrance
  2. 2.UMR CNRS EPOCBordeaux UniversityTalenceFrance

Personalised recommendations