Journal of Applied Mathematics and Computing

, Volume 59, Issue 1–2, pp 489–516

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

• Alia Al-Ghosoun
• Michael Herty
• Mohammed Seaid
Original Research

## Abstract

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.

## Keywords

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

## Mathematics Subject Classification

65N08 35L50 76M12 65J15

## Notes

### Acknowledgements

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).

## References

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)
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)
3. 3.
Benkhaldoun, F., Seaid, M.: A simple finite volume method for the shallow water equations. J. Comput. Appl. Math. 234, 58–72 (2010)
4. 4.
Bermudez, A., Vázquez-Cendón, M.E.: Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994)
5. 5.
Birman, J., Falcovitz, A.: Application of the GRP scheme to open channel flow equations. J. Comput. Phys. 222, 131–154 (2007)
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)
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)
8. 8.
Bouchut, F.: Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser, Basel (2004)
9. 9.
Brocchini, M., Dodd, N.: Nonlinear shallow water equations modeling for coastal engineering. J. Waterw. Port Coast. Ocean Eng. 134, 104–120 (2008)
10. 10.
Chen, S., Noelle, G.: A new hydrostatic reconstruction scheme based on subcell reconstructions. SIAM J. Numer. Anal. 55, 758–784 (2017)
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)
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)
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)
15. 15.
LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
16. 16.
LeVeque, R.J.: Numerical Methods for Conservation Laws. Lectures in Mathematics. ETH Zürich, Zürich (1992)
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)
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)
19. 19.
Li, Y., Raichlen, F.: Non-breaking and breaking solitary wave run-up. J. Fluid Mech. 456, 295–318 (2002)
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)
21. 21.
Roe, P.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43, 357–372 (1981)
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)
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)
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)
27. 27.
Vreugdenhil, C.B.: Numerical Method for Shallow Water Flow. Kluwer Academic, Dordsecht (1994)
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)
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)

© 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