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On a 2D ‘zoom’ for the 1D shallow water model: coupling and data assimilation. (English) Zbl 1173.76309
Summary: In the context of river hydraulics we elaborate the idea of a ‘zoom’ model locally superposed on an open-channel network global model. The zoom model (2D shallow water equations) describes additional physical phenomena, which are not represented by the global model (1D shallow water equations with storage areas). Both models are coupled using the optimal control approach when the zoom model is used to assimilate local observations into the global model (variational data assimilation) by playing the part of a mapping operator. The global model benefits from using zooms, while no substantial modification to it is required. Numerical results on a toy test case show the feasibility of the suggested method.

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76M30 Variational methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
Full Text: DOI
[1] Alifanov, O.M.; Artyukhin, E.A.; Rumyantsev, S.V., Extreme methods for solving ill-posed problems with applications to inverse heat transfer problems, (1996), Begel House Publishers · Zbl 1006.35001
[2] Belanger, E.; Vincent, A., Data assimilation (4DVAR) to forecast flood in shallow waters with sediment erosion, J. hydrol., 300, 14, 114125, (2005)
[3] Blayo, E.; Debreu, L., Revisiting open boundary conditions from the point of view characteristic variables, Ocean modell., 9, 231-252, (2005)
[4] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. comput., 31, 333-390, (1977) · Zbl 0373.65054
[5] Cunge, J.A.; Holly, F.M.; Verwey, A., Practical aspects of computational river hydraulics, (1980), Pitman London
[6] Formaggia, L.; Gerbeau, J.F.; Nobile, F.; Quarteroni, A., On the coupling of 3D and 1D navier – stokes equations for flow problems in compliant vessels, Comput. methods appl. mech. engrg., 191, 561-582, (2001) · Zbl 1007.74035
[7] Fox, A.D.; Maskell, S.J., Two-way interactive nesting of primitive equation Ocean models with topography, J. phys. oceanogr., 25, 2977-2996, (1995)
[8] M.J. Gander, L. Halpern, F. Nataf, Optimal convergence for overlapping and non-overlapping Schwarz waveform relaxation, in: C-H. Lai, P. Bjorstad, M. Cross, O. Widlund (Eds.), Eleventh International Conference on Domain Decomposition Methods, 1999, pp. 27-36.
[9] I. Gejadze, M. Honnorat, X. Lai, J. Marin, J. Monnier, Dassflow v2.0: a variational data assimilation software for river flows. Research Report INRIA, February 2007. http://ljk.imag.fr/MOISE/dassflow.
[10] Gejadze, I.Yu.; Copeland, G.J.M., Open boundary control problem for navier – stokes equations including a free surface: adjoint sensitivity analysis, Comput. math. appl., 52, 1269-1288, (2006) · Zbl 1118.49017
[11] Gervasio, P.; Lions, J.L.; Quarteroni, A., Heterogeneous coupling by virtual control methods, Numer. math., 90, 241-264, (2001) · Zbl 1002.65133
[12] Gilbert, J.Ch.; Lemarechal, C., Some numerical experiments with variable storage quasi-Newton algorithms, Math. programm., 45, 503-528, (1989)
[13] A. Hascoet, V. Pascual, TAPENADE 2.1 user’s guide, Technical report RT-300, INRIA, 2004.
[14] M. Honnorat, J. Monnier, F.-X. Le Dimet, Lagrangian data assimilation for river hydraulics simulations, Comp. Vis. Sci., in press. · Zbl 1426.86005
[15] E. Lelarasmee, The waveform relaxation method for the time domain analysis of large scale nonlinear dynamical systems, Ph.D. thesis, University of California, Berkeley, 1982.
[16] F.-X. Le Dimet, C. Mazauric, W. Castaings, Prospects for the use of data assimilation for flood prediction Workshop on Flood Prevention and Control on the Yangtze River (FOCYR), 2004.
[17] E. Miglio, S. Perotto, F. Saleri, Model coupling techniques for free-surface flow problems, Part I, in: Proceedings of the Fourth World Congress of Nonlinear Analysis WCNA-2004, Orlando, FL, USA, 2004. · Zbl 1224.76098
[18] Morozov, V.A., Methods for solving incorrectly posed problems, (1984), Springer
[19] Toro, E.F., Shock-capturing methods for free-surface shallow flows, (2001), John Wiley and Sons · Zbl 0996.76003
[20] Le Veque, R.J., Balancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave propagation algorithm, J. comp. phys., 146, 346-365, (1998) · Zbl 0931.76059
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