Numerical Scheme for a Viscous Shallow Water System Including New Friction Laws of Second Order: Validation and Application

  • Olivier Delestre
  • Ulrich RazafisonEmail author
Part of the Springer Water book series (SPWA)


In this work, we are interested in the derivation of a new shallow water model with a diffusion source term. Analytical solutions for steady flow regimes are first presented to validate a numerical method designed to solve this new model. Then this model is applied on real data and seems to give better results than the classical shallow water system.


Shallow water system Model derivation Finite volume scheme Well-balanced method Hydrostatic reconstruction Friction law Manning friction law Darcy-Weisbach friction law 



The authors whish to thanks the ANR-11-JS01-006-01 project CoToCoLa (Contemporary Topics on Conservation Laws), Carine Lucas for her advices and Frédéric Darboux for the data used in Sect. 4.2.


  1. 1.
    de Saint-Venant, A. J.-C. (1871). Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à l’introduction des marées dans leur lit. Comptes Rendus de l’Académie des Sciences, 73, 147–154.zbMATHGoogle Scholar
  2. 2.
    Delestre, O. (2010). Simulation du ruissellement d’eau de pluie sur des surfaces agricoles. PhD thesis University of Orléans, in french.
  3. 3.
    Delestre, O., Cordier, S., Darboux, F., Du, M., James, F., & Laguerre, C., et al. (2014). FullSWOF: A software for overland flow simulation. In P.Gourbesville, J. Cunge, & G. Caignaert, (Eds.), Advances in Hydroinformatics, Springer Hydrogeology (pp. 221–231). Springer: Singapore.Google Scholar
  4. 4.
    Esteves, M., Faucher, X., Galle, S., & Vauclin, M. (2000). Overland flow and infiltration modelling for small plots during unsteady rain: numerical results versus observed values. Journal of Hydrology, 228, 265–282.CrossRefGoogle Scholar
  5. 5.
    Tatard, L., Planchon, O., Wainwright, J., Nord, G., Favis-Mortlock, D., Silvera, N., et al. (2008). Measurement and modelling of high-resolution flow-velocity data under simulated rainfall on a low-slope sandy soil. Journal of Hydrology, 348(1–2), 1–12.CrossRefGoogle Scholar
  6. 6.
    Goutal, N., & Maurel, F. (2002). A finite volume solver for 1D shallow-water equations applied to an actual river. International Journal for Numerical Methods in Fluids, 38, 1–19.CrossRefzbMATHGoogle Scholar
  7. 7.
    Caleffi, V., Valiani, A., & Zanni, A. (2003). Finite volume method for simulating extreme flood events in natural flood events in natural channels. Journal of Hydraulic Research, 41(2), 167–177.CrossRefGoogle Scholar
  8. 8.
    Alcrudo, F., & Gil, E. (1999). The Malpasset dam break case study. Proceedings of the 4th CADAM Workshop, Zaragoza (pp. 95–109).Google Scholar
  9. 9.
    Valiani, A., Caleffi, V., & Zanni, A. (2002). Case study: Malpasset dam-break simulation using a two-dimensional finite volume methods. Journal of Hydraulic Engineering, 128(5), 460–472.CrossRefGoogle Scholar
  10. 10.
    Popinet, S. (2011). Quadtree-adaptive tsunami modelling. Ocean Dynamics, 61(9), 1261–1285.CrossRefGoogle Scholar
  11. 11.
    Gerbeau, J.-F., & Perthame, B. (2001). Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete and Continuous Dynamical Systems---Series S, 1, 89–102.CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Marche, F. (2007). Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects. European Journal of Mechanics B/Fluids, 26, 49–63.CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    MacDonald, I., Baines, M. J., Nichols, N. K., & Samuels, P. G. (1997). Journal of Hydraulic Engineering, 123, 1041–1045.CrossRefGoogle Scholar
  14. 14.
    Chow, V. T. (1959). Open-channel hydraulics. New York: McGraw-Hill.Google Scholar
  15. 15.
    Delestre, O., Darboux, F., James, F., Lucas, C., Laguerre, C., & Cordier, S. (Submitted). FullSWOF: A free software package for the simulation of shallow water flows. Scholar
  16. 16.
    Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., & Perthame, B. (2004). A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. Journal of Scientific Computing, 25(6), 2050–2065.CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Bouchut, F. (2004). Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. Frontiers in Mathematics. Basel: Birkhauser.Google Scholar
  18. 18.
    Bristeau, M.-O., & Coussin, B. (2001). Boundary conditions for the shallow water equations solved by kinetic schemes. Inria report RR-4282.Google Scholar
  19. 19.
    Fiedler, R. F., & Ramirez, J. A. (2000). A numerical method for simulating discontinuous shallow flow over an infiltrating surface. International Journal for Numerical Methods in Fluids, 32, 219–240.CrossRefzbMATHGoogle Scholar
  20. 20.
    Anderson, E., Bai, Z., Bischof, C., Blackford, L.S., Demmel, J., & Dongarra, J., et al. (1999). LAPACK Users’ guide (3rd ed.). Philadelphia: Society for Industrial and Applied Mathematics.Google Scholar
  21. 21.
    Delestre, O., Lucas, C., Ksinant, P.-A., Darboux, F., Laguerre, C., Vo, T. N. T., et al. (2013). SWASHES: A compilation of Shallow-Water analytic solutions for hydraulic and environmental studies. International Journal for Numerical Methods in Fluids, 72, 269–300. doi: 10.1002/fld.3741.CrossRefMathSciNetGoogle Scholar
  22. 22.
    Delestre, O., Lucas, C., Ksinant, P.-A., Darboux, F., Laguerre, C., & James, F., et al. (2014). SWASHES: A library for benchmarking in hydraulic. In p. Gourbesville, J. Cunge & G. Caignaert, (Eds.), Advances in Hydroinformatics, Springer Hydrogeology (pp. 233–243). Springer: Singapore.Google Scholar
  23. 23.
    Legout, C., Darboux, F., Nédélec, Y., Hauet, A., Esteves, M., Renaux, B., et al. (2012). High spatial resolution mapping of surface velocities and depths for shallow overland flow. Earth Surface Processes and Landforms, 37(9), 984–993. doi: 10.1002/esp.3220.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Laboratory J.A. Dieudonné CNRS UMR 7351 & Polytech Nice – SophiaUNSAArequipaPeru
  2. 2.Laboratoire de Mathématiques, CNRS UMR 6623Université de Franche-ComtéBesançon CedexFrance

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