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

Two-Scale numerical simulation of sand transport problems

Ibrahima Faye 1, , Emmanuel Frénod 2, and  Diaraf Seck 3,

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.
Keywords: long time behavior, Homogenization, asymptotic expansion, asymptotic analysis, dune and megaripple morphodynamics, modeling coastal zone phenomena, numerical simulation..
Mathematics Subject Classification: Primary: 35K65, 35B25, 35B10; Secondary: 92F05, 86A6.
Citation: Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Thèse de doctorat, Strasbourg, 2009.

[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.

[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.

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Thèse de doctorat, Strasbourg, 2009.

[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.

[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.

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