Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 489–516 | Cite as

A new numerical treatment of moving wet/dry fronts in dam-break flows

  • Alia Al-GhosounEmail author
  • Michael Herty
  • Mohammed Seaid
Original Research


The aim of this paper is to present a new finite volume method for moving wet/dry fronts in shallow water flows. The method consists on reformulating the shallow water equations in a moving wetted domain where the wet/dry interface is located using the speed of the water flow. A set of parametrized coordinates is introduced and the underlying equations are transformed to a new hyperbolic system with advection terms to be solved in fixed domains. A well-balanced finite volume method is developed to approximate numerical solutions of the parametrized system. We derive a well-balanced approximation of the source terms and prove that the proposed method is well-balanced for the shallow water flows in the presence of moving wet/dry fronts over non-flat topography. Several numerical results confirm the reliability and accuracy of the new method.


Shallow water equations Wet/dry fronts Finite volume method Well-balanced discretization Dam-break problems 

Mathematics Subject Classification

65N08 35L50 76M12 65J15 



This work has been supported by BMBF KinOpt 05M2013 and DFG Cluster of Excellence EXC128. The work of M. Seaid was supported in part by Deutscher Akademischer Austauschdienst (DAAD).


  1. 1.
    Audusse, E., Bouchut, F., Bristeau, M.O., Klein, R., Perthame, B.: A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Benkhaldoun, F., Sari, S., Seaid, M.: A family of finite volume Eulerian–Lagrangian methods for two-dimensional conservation laws. J. Comput. Appl. Math. 285, 181–202 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Benkhaldoun, F., Seaid, M.: A simple finite volume method for the shallow water equations. J. Comput. Appl. Math. 234, 58–72 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bermudez, A., Vázquez-Cendón, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Birman, J., Falcovitz, A.: Application of the GRP scheme to open channel flow equations. J. Comput. Phys. 222, 131–154 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bollermann, A., Chen, G., Kurganov, A., Noelle, S.: A well-balanced reconstruction of wet/dry fronts for the shallow water equations. J. Sci. Comput. 56, 267–290 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bollermann, A., Noelle, S., Lukácová-Medvidová, M.: Finite volume evolution Galerkin methods for the shallow water equations with dry beds. Commun. Comput. Phys. 10, 371–404 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser, Basel (2004)CrossRefzbMATHGoogle Scholar
  9. 9.
    Brocchini, M., Dodd, N.: Nonlinear shallow water equations modeling for coastal engineering. J. Waterw. Port Coast. Ocean Eng. 134, 104–120 (2008)CrossRefGoogle Scholar
  10. 10.
    Chen, S., Noelle, G.: A new hydrostatic reconstruction scheme based on subcell reconstructions. SIAM J. Numer. Anal. 55, 758–784 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    de Saint-Venant, A.J.C.: Théorie du mouvement non permanent des eaux, avec application aux crues des riviére at á l’introduction des warées dans leurs lits. Comptes Rendus des séances de l’Académie des Sciences 73, 237–240 (1871)Google Scholar
  12. 12.
    Ern, A., Piperno, S., Djade, K.: A well-balanced Runge–Kutta discontinuous Galerkin method for the shallow water equations with flooding and drying. Int. J. Numer. Methods Fluids 58, 1–25 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gallardo, J.M., Parés, C., Castro, M.: On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas. J. Comput. Phys. 227, 574–601 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Glimm, J., Marshall, G., Plohr, B.: A generalized Riemann problem for quasi-one-dimensional gas flows. Adv. Appl. Math. 5, 1–30 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)CrossRefGoogle Scholar
  16. 16.
    LeVeque, R.J.: Numerical Methods for Conservation Laws. Lectures in Mathematics. ETH Zürich, Zürich (1992)CrossRefzbMATHGoogle Scholar
  17. 17.
    LeVeque, R.J.: Balancing source terms and fux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm. J. Comput. Phys. 146, 346–365 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, G., Chen, J.: The generalized Riemann problem method for the shallow water equations with bottom topography. J. Numer. Methods Eng. 65, 834–862 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, Y., Raichlen, F.: Non-breaking and breaking solitary wave run-up. J. Fluid Mech. 456, 295–318 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Noelle, S., Pankratz, N., Puppo, G., Natvig, J.R.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Roe, P.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Stoker, J.J.: Water Waves. Interscience Publishers Inc, New York (1986)Google Scholar
  23. 23.
    Temperton, C., Staniforth, A.: An efficient two-time-level semi-Lagrangian semi-implicit integration scheme. Q. J. R. Meteorol. Soc. 113, 1025–1039 (1987)CrossRefGoogle Scholar
  24. 24.
    Titov, V.V., Synolakis, C.E.: Numerical modeling of tidal wave run-up. J. Waterw. Port Coast. Ocean Eng. 124, 157–171 (1998)CrossRefGoogle Scholar
  25. 25.
    Toro, E.F.: The dry-bed problem in shallow-water flows. Technical report no. 9007. College of Aeronautics Reports (1990)Google Scholar
  26. 26.
    Toro, E.F.: Shock-Capturing Methods for Free-Surface Shallow Flows. Wiley, New York (2002)zbMATHGoogle Scholar
  27. 27.
    Vreugdenhil, C.B.: Numerical Method for Shallow Water Flow. Kluwer Academic, Dordsecht (1994)CrossRefzbMATHGoogle Scholar
  28. 28.
    Xing, Y., Shu, C.W.: High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 3206–227 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhou, F., Chen, G., Huang, Y., Feng, H.: An adaptive moving finite volume scheme for modeling flood inundation over dry and complex topography. Water Resour. Res. 49, 1914–1928 (2013)CrossRefGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  • Alia Al-Ghosoun
    • 1
    Email author
  • Michael Herty
    • 2
  • Mohammed Seaid
    • 1
  1. 1.School of Engineering and Computing SciencesUniversity of DurhamDurhamUK
  2. 2.IGPMRWTH Aachen UniversityAachenGermany

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