American Institute of Mathematical Sciences

Advanced
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 & 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

show all references

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

[1]

Kun Wang, Yinnian He, Yanping Lin. Long time numerical stability and asymptotic analysis for the viscoelastic Oldroyd flows. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1551-1573. doi: 10.3934/dcdsb.2012.17.1551
[2]

Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199
[3]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 509-528. doi: 10.3934/dcds.2007.17.509
[4]

Jiaohui Xu, Tomás Caraballo, José Valero. Asymptotic behavior of nonlocal partial differential equations with long time memory. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021140
[5]

Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205
[6]

Min Chen, Olivier Goubet. Long-time asymptotic behavior of two-dimensional dissipative Boussinesq systems. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 37-53. doi: 10.3934/dcdss.2009.2.37
[7]

Chaoying Li, Xiaojing Xu, Zhuan Ye. On long-time asymptotic behavior for solutions to 2D temperature-dependent tropical climate model. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021163
[8]

Long Wei. Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions. Communications on Pure & Applied Analysis, 2008, 7 (4) : 925-946. doi: 10.3934/cpaa.2008.7.925
[9]

Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure & Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347
[10]

Jean-Paul Chehab. Damping, stabilization, and numerical filtering for the modeling and the simulation of time dependent PDEs. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2693-2728. doi: 10.3934/dcdss.2021002
[11]

Philip M. J. Trevelyan. Approximating the large time asymptotic reaction zone solution for fractional order kinetics $A^n B^m$. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 219-234. doi: 10.3934/dcdss.2012.5.219
[12]

Yue Qiu, Sara Grundel, Martin Stoll, Peter Benner. Efficient numerical methods for gas network modeling and simulation. Networks & Heterogeneous Media, 2020, 15 (4) : 653-679. doi: 10.3934/nhm.2020018
[13]

Vincenzo Michael Isaia. Numerical simulation of universal finite time behavior for parabolic IVP via geometric renormalization group. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3459-3481. doi: 10.3934/dcdsb.2017175
[14]

Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023
[15]

Thierry Paul, Mario Pulvirenti. Asymptotic expansion of the mean-field approximation. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1891-1921. doi: 10.3934/dcds.2019080
[16]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258
[17]

Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47
[18]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609
[19]

Nakao Hayashi, Pavel I. Naumkin. Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation revisited. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 383-400. doi: 10.3934/dcds.1997.3.383
[20]

Tingting Liu, Qiaozhen Ma. Time-dependent asymptotic behavior of the solution for plate equations with linear memory. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4595-4616. doi: 10.3934/dcdsb.2018178

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (71)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy