Jean De Dieu Zabsonre - Academia.edu (original) (raw)

Papers by Jean De Dieu Zabsonre

DOAJ (DOAJ: Directory of Open Access Journals), Jul 1, 2017

The existence of global weak solutions in a periodic domain for a non-linear viscous bi-layer sha... more The existence of global weak solutions in a periodic domain for a non-linear viscous bi-layer shallow-water model with capillarity effects and extra friction terms in a two-dimensional space has been proved in [21]. The main contribution of this article is to show the existence of global weak solutions without friction term or capillary effect following the ideas of [20] for the two dimensional case.

Annals of the University of Craiova - Mathematics and Computer Science Series, Jun 15, 2012

In this paper we are concerned with the stability of weak solutions for a family of two-dimension... more In this paper we are concerned with the stability of weak solutions for a family of two-dimensional bed-load transport models which combines a viscous shallow water system with a transport equation that describes the bottom evolution. Our analysis is performed in a periodic domain where models with critical shear stress are used for the solid discharge.

Cmes-computer Modeling in Engineering & Sciences, Mar 1, 2009

In this work we present a new two-dimensional bilayer Shallow-Water model including viscosity and... more In this work we present a new two-dimensional bilayer Shallow-Water model including viscosity and friction effects on the bottom and interface level. It is obtained following [[6]] from an asymptotic analysis of non-dimensional and incompressible Navier-Stokes equations with hydrostatic approximation. In order to obtain the viscosity effects into the model we must have into account a second order approximation. To evaluate this model we perform two numerical tests consisting of an internal dam-break problem for both, one and two dimensional cases. In the first one we make a comparison between the model obtained and the Navier-Stokes simulation. Shallow Water equations, bilayer models, viscosity, friction, capillarity, Finite Volume methods.

Communications in Mathematical Sciences, 2012

The aim of this article is to derive a simplified sedimentation model thanks to an asymptotic ana... more The aim of this article is to derive a simplified sedimentation model thanks to an asymptotic analysis. We consider a two time scale erosion process due to tidal effects and we show that the approximation at the first order can model bed-load transport well. To this end, the simplified model is validated through numerical tests (evolution of a dune submitted to tidal effects in the ocean, run up near the coast) and compared to direct simulations that are very expensive in terms of computation time.

Nonlinear Analysis-real World Applications, Oct 1, 2009

We consider a non-linear viscous bi-layer shallow water model with capillarity effects and extra ... more We consider a non-linear viscous bi-layer shallow water model with capillarity effects and extra friction terms in a two-dimensional space. This system is issued from a derivation of a three-dimensional Navier-Stokes equations with water-depth depending on friction coefficients. We prove an existence result for global weak solution in a periodic domain Ω = T 2 .

HAL (Le Centre pour la Communication Scientifique Directe), May 4, 2019

This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes... more This work is devoted to the derivation of an energy estimate to be satisfied by numerical schemes when approximating the weak solutions of the shallow water model. More precisely, here we adopt the well-known hydrostatic reconstruction technique to enforce the adopted scheme to be well-balanced; namely to exactly preserve the lake at rest stationary solution. Such a numerical approach is known to get a semi-discrete (continuous in time) entropy inequality. However, a semi-discrete energy estimation turns, in general, to be insufficient to claim the required stability. In the present work, we adopt the artificial numerical viscosity technique to increase the desired stability and then to recover a fully discrete energy estimate. Several numerical experiments illustrate the relevance of the designed viscous hydrostatic reconstruction scheme.

Nous presentons dans ce document des modeles d'ecoulements bicouches. Il s'agit de modele... more Nous presentons dans ce document des modeles d'ecoulements bicouches. Il s'agit de modeles d'ecoulement en eaux peu profondes et de modeles de transport de sediments. Nous derivons dans un premier temps des modeles de Saint-Venant visqueux, bicouches et bidimensionnels en supposant que l'ecoulement est compose de deux fluides immiscibles (cas du Detroit de Gibraltar). Nous donnons quelques resultats numeriques sur les modeles visqueux derives. On etend alors les resultats d'existence de solutions obtenus dans le cas monocouche au cas bicouches. Dans cette analyse, la difficulte provient des termes de frottement au vu des multiplicateurs utilises dans les estimations d'entropies. Nous proposons ensuite de nouveaux modeles de transport de sediments energetiquement consistants pour lesquels nous obtenons des resultats theoriques de stabilite. Enfin, nous developpons une nouvelle version flux limiteur de schema numerique volumes finis, bien equilibre, en combinant un schema de type roe et de Lax-Wendroff, tous deux etant construits en tenant compte de la variation tangentielle des quantites. Ce schema numerique est utilise pour simuler le transport de sediment.

Asymptotic Analysis, Sep 7, 2016

This paper is devoted to the derivation and the study of a two-dimensional bilayer model. This mo... more This paper is devoted to the derivation and the study of a two-dimensional bilayer model. This model is obtained from the incompressible Navier-Stokes equations with suitable boundary conditions including friction and capillary effects as in the method used by [European J. Applied Mathematics 24(6) (2013), 803-833] in the one dimensional case. We perform a multiscale analysis in space and time, as well as an asymptotic analysis to obtain a system coupling two different features: the Reynolds lubrication equations for the upper layer and the shallow water equation for the lower one. We finally prove the existence of global weak solutions in time for model containing some additional terms.

European Journal of Applied Mathematics, Jun 27, 2013

In this paper a bilayer model is derived to simulate the evolution of a thin film flow over water... more In this paper a bilayer model is derived to simulate the evolution of a thin film flow over water. This model is derived from the incompressible Navier-Stokes equations together with suitable boundary conditions including friction and capillary effects. The derivation is based on the different properties of the fluids, thus, we perform a multiscale analysis in space and time, and a different asymptotic analysis to derive a system coupling two different models: the Reynolds lubrication equation for the upper layer and the shallow water model for the lower one. We prove that the model is provided of a dissipative entropy inequality, up to a second order term. Moreover, we propose a correction of the model −by taking into account the second order extention for the pressure− that admits an exact dissipative entropy inequality. Two numerical tests are presented. In the first one we compare the numerical results with the viscous bilayer shallow water model proposed in [G. Narbona-Reina, J.D.D. Zabsonré, E.D. Fernández-Nieto, D. Bresch, CMES Comput. Model. Eng. Sci., 2009]. In the second test the objective is to show some of the characteristic situations that can be studied with the proposed model. We simulate a problem of pollutant dispersion near the coast. For this test the influence of the friction coefficient on the coastal area affected by the pollutant is studied.

Mathematical Modelling and Numerical Analysis, Nov 28, 2016

In this work we present a deduction of the Saint-Venant-Exner model through an asymptotic analysi... more In this work we present a deduction of the Saint-Venant-Exner model through an asymptotic analysis of the Navier-Stokes equations. A multi-scale analysis is performed in order to take into account that the velocity of the sediment layer is smaller than the one of the fluid layer. This leads us to consider a shallow water type system for the fluid layer and a lubrication Reynolds equation for the sediment one. This deduction provides some improvements with respect to the classical Saint-Venant-Exner model: (i) the deduced model has an associated energy. Moreover, it allows us to explain why classical models do not have an associated energy and how to modify them in order to recover a model with this property. (ii) The model incorporates naturally a necessary modification that must be taken into account in order to be applied to arbitrarily sloping beds. Furthermore, we show that this modification is different of the ones considered classically, and that it coincides with a classical one only if the solution has a constant free surface. (iii) The deduced solid transport discharge naturally depends on the thickness of the moving sediment layer, what allows to ensure sediment mass conservation. Moreover, we include a simplified version of the model for the case of quasi-stationary regimes. Some of these simplified models correspond to the generalization of classical ones such as Meyer-Peter&Müller and Ashida-Michiue models. Three numerical tests are presented to study the evolution of a dune for several definition of the repose angle, to see the influence of the proposed definition of the effective shear stress in comparison with the classical one, and by comparing with experimental data.

Nonautonomous Dynamical Systems

In the abstract, homogenize the references as follows: In this paper,we study the existence of gl... more In the abstract, homogenize the references as follows: In this paper,we study the existence of global weak solutions of a two dimensionnal model. The model is inspired by the one studied in [Math. Models Methods Appl. Sci. 19 (2009), 477-499]. Our model is a particular case of the two-dimensionnal sediment transport model derived in [Mathematical Modelling and Numerical Analysis 51 (2017), 115-145]. The main contribution of this article is to extend the results obtained 10 in [C. R. Math. Acad. Sci. Paris, 344 (2007), 443-446; Math. Models Methods Appl. Sci. 19 (2009), 477-499] by showing the existence of global weak solution of a sediment transport model without assuming the data to be small enough, following the ideas proposed in [Inv. Math.206 (2016), 935-974].

Annals of the University of Craiova - Mathematics and Computer Science Series, 2012

The aim of this paper is to prove the uniqueness of strong solutionof a one dimensional viscous b... more The aim of this paper is to prove the uniqueness of strong solutionof a one dimensional viscous bilayer shallow water model. Our analy-sis is based on some new useful estimate namely BD entropy and on amethod developed by Mellet and Vasseur in [14] to prove the existenceand uniqueness on some compressible one dimensional Navier-Stokessystem. Under suitable assumptions on the solutions and using Gron-wall Lemma, we obtain the uniqueness of strong solution. We performour analysis in periodic domain with periodic boundaries conditions.

Annals of the University of Craiova - Mathematics and Computer Science Series, 2017

We consider a one-dimensional bilayer model coupling shallow water and Reynolds lubrication equat... more We consider a one-dimensional bilayer model coupling shallow water and Reynolds lubrication equations that is a similar model to [European J. Applied Mathematics 24(6) (2013), 803-833]. The model considered is represented by the two superposed immiscible uids. Under a physical hypothesis, we show the existence of global weak solution in time in periodic domain with periodic boundaries conditions.

Comptes Rendus Mathematique, 2011

We consider a system composed by two immiscible fluids in two-dimensional space that can be model... more We consider a system composed by two immiscible fluids in two-dimensional space that can be modelized by a bilayer Shallow Water equations with extra friction terms and capillary effects. We give an existence theorem of global weak solutions in a periodic domain. Résumé Nous considérons un système composé par deux fluides immiscibles dans un domaine bi-dimensionnel pouvantêtre représenté par un modèle bicouche visqueux de Saint-Venant avec des termes de friction additionnels et des effets de capillarité. Nous donnons un théorème d'existence de solutions faibles globales dans un domaine périodique. Version française abrégée Dans cette note, nous nous intéressonsà l'étude de l'existence de solutions faibles globales en temps d'un modèle bicouche visqueux de Saint-Venant dérivé dans [6]. Notons que dans le cas d'une couche, dans [1] et [4] les auteurs ont obtenu l'existence de solutions faibles grâceà une nouvelle entropie (BD) introduite premièrement par Bresch et Desjardins dans [1]. On peut trouver d'autres résultats sur l'existence de solutions pour des modèles bicouche de Saint-Venant dans [3] et [5]. Dans ces modèles, les termes couplant les deux fluides compliquent le passageà la limite. Dans [7] uneétude du modèle bicouche mais où les termes de friction ontété simplifiés aété faite. Les termes de friction couplant les deux couches dans le

Journal of Scientific Computing, 2019

Journal of Mathematics Research, 2017

We consider a one-dimensionnal bilayer model coupling shallow water and Reynolds lubrication equa... more We consider a one-dimensionnal bilayer model coupling shallow water and Reynolds lubrication equations with a molecular interactions between molecules. These molecular interactions give rise to intermolecular forces, namely the long-range van der Waals forces and short-range Born intermolecular forces. In this paper, an expression will be used to take into account all these intermolecular forces. Our model is a similar model studied in (Roamba, Zabsonré & Zongo, 2017). The model considered is represented by the two superposed immiscible fluids. A similar model was studied in (Zabsonré Lucas & Fernandez-Nieto, 2009) but the authors do not take into account the intermolecular forces. Without hypothesis about the unknowns as in (Roamba, Zabsonré & Zongo, 2017), we show the existence of global weak solution in time in a periodic domain.

Portugaliae Mathematica, 2012

Nonlinear Analysis: Real World Applications, 2013

In this paper, we consider a viscous bilayer shallow water model in one space dimension that repr... more In this paper, we consider a viscous bilayer shallow water model in one space dimension that represents two superposed immiscible fluids. For this model, we prove the existence of strong solutions in a periodic domain. The initial heights are required to be bounded above and below away from zero and we get the same bounds for every time. Our analysis is based on the construction of approximate systems which satisfy the BD entropy and on the method developed by A. Mellet and A. Vasseur to obtain the existence of global strong solutions for the one dimensional Navier-Stokes equations.

Mediterranean Journal of Mathematics, 2013

ABSTRACT

Mathematical Models and Methods in Applied Sciences, 2009

In this paper we consider a two-dimensional viscous sedimentation model which is a viscous Shallo... more In this paper we consider a two-dimensional viscous sedimentation model which is a viscous Shallow–Water system coupled with a diffusive equation that describes the evolution of the bottom. For this model, we prove the stability of weak solutions for periodic domains and give some numerical experiments. We also discuss around various discharge quantity choices.