# American Institute of Mathematical Sciences

February  2015, 8(1): 151-168. doi: 10.3934/dcdss.2015.8.151

## Two-Scale numerical simulation of sand transport problems

 1 Université Alioune Diop de Bambey, UFR S.A.T.I.C, BP 30 Bambey (Sénégal), Ecole Doctorale de Mathématiques et Informatique, Laboratoire de Mathématiques de la Décision et d'Analyse Numérique, (L.M.D.A.N) F.A.S.E.G)/F.S.T., Senegal 2 Université de Bretagne-Sud, LMBA - UMR6205, Centre Yves Coppens, Campus de Tohannic, F-56017, Vannes Cedex, France 3 Université Cheikh Anta Diop de Dakar, BP 16889 Dakar Fann, Ecole Doctorale de Mathématiques et Informatique, Laboratoire de Mathématiques de la Décision et d'Analyse Numérique, (L.M.D.A.N) F.A.S.E.G, Senegal

Received  April 2013 Revised  September 2013 Published  July 2014

In this paper we consider the model built in [3] for short term dynamics of dunes in tidal area. We construct a Two-Scale Numerical Method based on the fact that the solution of the equation which has oscillations Two-Scale converges to the solution of a well-posed problem. This numerical method uses on Fourier series.
Citation: Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar [2] P. Aillot, E. Frénod and V. Monbet, Long term object drift in the ocean with tide and wind, Multiscale Model. and Simul., 5 (2006), 514-531 (electronic). doi: 10.1137/050639727.  Google Scholar [3] I. Faye, E. Frénod and D. Seck, Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment, Discrete Contin. Dyn. Syst., 29 (2011), 1001-1030. doi: 10.3934/dcds.2011.29.1001.  Google Scholar [4] E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. of Pure Appl. Math. Adv. Appl., 4 (2010), 135-169.  Google Scholar [5] E. Frénod, A. Mouton and E. Sonnendrücker, Two-Scale numerical simulation of the weakly compressible 1D isentropic Euler equations, Numer. Math., 108 (2007), 263-293. doi: 10.1007/s00211-007-0116-8.  Google Scholar [6] E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci., 19 (2009), 175-197. doi: 10.1142/S0218202509003395.  Google Scholar [7] E. Frénod, P. A. Raviart and E. Sonnendrücker, Two scale expansion of a singularly perturbed convection equation, J. Math. Pures Appl. (9), 80 (2001), 815-843. doi: 10.1016/S0021-7824(01)01215-6.  Google Scholar [8] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, (Russian) Translated from the Russian by S. Smith, Translation of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.  Google Scholar [9] A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Thèse de doctorat, Strasbourg, 2009.  Google Scholar [10] A. Mouton, Two-Scale semi-Lagrangian simulation of a charged particule beam in a periodic focusing channel, Kinet. Relat. Models, 2 (2009), 251-274. doi: 10.3934/krm.2009.2.251.  Google Scholar [11] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.  Google Scholar [2] P. Aillot, E. Frénod and V. Monbet, Long term object drift in the ocean with tide and wind, Multiscale Model. and Simul., 5 (2006), 514-531 (electronic). doi: 10.1137/050639727.  Google Scholar [3] I. Faye, E. Frénod and D. Seck, Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment, Discrete Contin. Dyn. Syst., 29 (2011), 1001-1030. doi: 10.3934/dcds.2011.29.1001.  Google Scholar [4] E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. of Pure Appl. Math. Adv. Appl., 4 (2010), 135-169.  Google Scholar [5] E. Frénod, A. Mouton and E. Sonnendrücker, Two-Scale numerical simulation of the weakly compressible 1D isentropic Euler equations, Numer. Math., 108 (2007), 263-293. doi: 10.1007/s00211-007-0116-8.  Google Scholar [6] E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci., 19 (2009), 175-197. doi: 10.1142/S0218202509003395.  Google Scholar [7] E. Frénod, P. A. Raviart and E. Sonnendrücker, Two scale expansion of a singularly perturbed convection equation, J. Math. Pures Appl. (9), 80 (2001), 815-843. doi: 10.1016/S0021-7824(01)01215-6.  Google Scholar [8] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, (Russian) Translated from the Russian by S. Smith, Translation of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968.  Google Scholar [9] A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Thèse de doctorat, Strasbourg, 2009.  Google Scholar [10] A. Mouton, Two-Scale semi-Lagrangian simulation of a charged particule beam in a periodic focusing channel, Kinet. Relat. Models, 2 (2009), 251-274. doi: 10.3934/krm.2009.2.251.  Google Scholar [11] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043.  Google Scholar
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