Abstract
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.
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References
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.
Delestre, O. (2010). Simulation du ruissellement d’eau de pluie sur des surfaces agricoles. PhD thesis University of Orléans, in french. http://tel.archives-ouvertes.fr/INSMI/tel-00531377/fr.
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.
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.
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.
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.
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.
Alcrudo, F., & Gil, E. (1999). The Malpasset dam break case study. Proceedings of the 4th CADAM Workshop, Zaragoza (pp. 95–109).
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.
Popinet, S. (2011). Quadtree-adaptive tsunami modelling. Ocean Dynamics, 61(9), 1261–1285.
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.
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.
MacDonald, I., Baines, M. J., Nichols, N. K., & Samuels, P. G. (1997). Journal of Hydraulic Engineering, 123, 1041–1045.
Chow, V. T. (1959). Open-channel hydraulics. New York: McGraw-Hill.
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. arxiv.org/abs/1401.4125.
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.
Bouchut, F. (2004). Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. Frontiers in Mathematics. Basel: Birkhauser.
Bristeau, M.-O., & Coussin, B. (2001). Boundary conditions for the shallow water equations solved by kinetic schemes. Inria report RR-4282.
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.
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.
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.
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.
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.
Acknowledgements
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.
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Delestre, O., Razafison, U. (2016). Numerical Scheme for a Viscous Shallow Water System Including New Friction Laws of Second Order: Validation and Application. In: Gourbesville, P., Cunge, J., Caignaert, G. (eds) Advances in Hydroinformatics. Springer Water. Springer, Singapore. https://doi.org/10.1007/978-981-287-615-7_16
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