Numerical Modelling and Simulation of Sand Dune Formation in an Incompressible Out-Flow

Yahaya Mahamane, Saley Bisso
DOI: 10.4236/am.2015.65080   PDF   HTML   XML   2,691 Downloads   3,300 Views  


In this paper, we are concerned with computation of a mathematical model of sand dune formation in a water of surface to incompressible out-flows in two space dimensions by using Chebyshev projection scheme. The mathematical model is formulate by coupling Navier-Stokes equations for the incompressible out-flows in 2D fluid domain and Prigozhin’s equation which describes the dynamic of sand dune in strong parameterized domain in such a way which is a subset of the fluid domain. In order to verify consistency of our approach, a relevant test problem is considered which will be compared with the numerical results given by our method.

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Nouri, Y. and Bisso, S. (2015) Numerical Modelling and Simulation of Sand Dune Formation in an Incompressible Out-Flow. Applied Mathematics, 6, 864-876. doi: 10.4236/am.2015.65080.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Prigozhin, L. (1994) Sandpiles and River Networks: Extended Systems with Nonlocal Interactions. Physical Review E, 49, 1161.
[2] Igbida, N. (2012) Mathematical Models for Sandpile Problems. XLIM-DMI, UMR-CNRS 6172, Workshop MathEnv, Essaouira.
[3] Nouri, Y.M. and Bisso, S. (2013) Numerical Approach for Solving a Mathematical Model of Sand Dune Formation. Pioneer Journal of Advances in Applied Mathematics, 9, 1-15.
[4] R?nquist, E.M. (1990) Optimal Spectral Element Methods for the Unsteady 3-Dimensionnal Incompressible Navier-Stokes Equations. Ph.D. Thesis, Mass, Cambridge.
[5] Azaez, M. (1990) Computation of the Pressure in the Stokes Problem for Incompressible Viscous Fluids by a Spectral Method Collocation. Thesis of Doctorate, Paris-Sud University, Orsay.
[6] Botella, O. (1996) Resolution des equations de Navier-Stokes par des schemas de Projection Tchebychev. Rapport de recherche No. 3018 de L’Institut national de rechercheen informatique et en automatique (inria).
[7] Maday, Y., Patera, A.T. and R?nquist, E.M. (1992) The Method for the Approximation of the Stokes Problem. Laboratoire d’Analyse Numerique, Paris VI, 11, fasc.4.
[8] Chorin, A. (1968) Numerical Simulation of the Navier-Stokes Equations. Mathematics of Computation, 22, 745-762.
[9] Temam, R. (1969) On the Approximation of the Solution of Navier-Stokes Equations by the Fractional Steps Method II. Archive for Rational Mechanics and Analysis, 32, 377-385.
[10] Azaez, M., Bernardi, C. and Grundmann, M. (1994) Spectral Methods Applied to Porous Media Equations. East-West Journal of Numerical Mathematics, 2, 91-105.
[11] Brezzi, F. (1974) On the Existence, Uniqueness and Approximation of Saddle-Point Problems Arising from Lagrangian Multipliers. R.A.I.R.O, R2, 129-151.
[12] Hesthaven, J.S., Gottlieb, S. and Gottlieb, D. (2007) Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge.
[13] Trefethen, L.N. (2000) Spectral Methods in MATLAB.
[14] Canuto, C., Bernardi, C. and Maday, Y. (1986) Generalized Inf-Sup Condition for Chebyshev Approximation of the Navier-Stokes Equations. Technical Report, No. 86-61, ICASE.
[15] Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (1988) Spectral Methods in Fluid Dynamics. Springer-Verlag, New York.
[16] Ehrenstein, U. and Peyret, R. (1989) A Chebyshev-Collocation Method for the Navier-Stokes Equations with Application to Double-Diffusive Convection. International Journal for Numerical Methods in Fluids, 9, 427-452.
[17] Azaez, M., Fikri, A. and Labrosse, G. (1994) A Unique Grid Spectral Solver of the nd Cartesian Unsteady Stokes System. Illustrative Numerical Results. Finite Elements in Analysis and Design, 16, 247-260.

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