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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.

The sandbank is a real physical phenomenon that constitutes a threat for our environment through the occu- pation of the roads, the arable earths and especially the waters of surfaces, as it is the case of the Niger stream. The main goal of this paper is to compute numerically the height of sand dune in a water of surface to the incompressible out-flows (streams, lakes, seas, ...). For this, we formulate a mathematical model which couples the Navier-Stokes equations for the incompressible out-flows in two space dimensions and Prigozhin’s equation that describes the sand dune dynamic [

The outline of this paper is as follows. In Section 2, we give the problem formulation and description of parameters. In Section 3, the numerical scheme which will be used in this paper is presented. In Section 4, some numerical simulations of the solution and temporal errors evolution are presented. We end this paper with a conclusion and the perspectives in Section 5.

Let

density is transported by a flux

action between the out-flow of the fluid and the dynamics of the dune in two space dimensions given by:

where

・

・

・

・

・

・

・ T is a given positive time-parameter.

The out-flow of the fluid is modelling by Equations (1)-(4). The transportation of the sand grains under the effect of averaged velocity is modelling by Equation (5). The dynamics of the sand dune is modelling by Equ- ations (6)-(8).

To ensure the regularity of the solution we suppose that the functions f,

and the initial data

where n is the unit vector normal to the boundary of the domain

For a given positif integer r, we consider a time step discretisation

knots of the interval

For a given continues function

In order to approach in time Equations (1)-(8), we used second-order backward Euler scheme which is given by:

While doing an extrapolation of order 1 of the pressure at the time of the prediction stage and while appro- aching the convection term

- prediction stage:

where

- projection stage:

where

Thus, when one does a spatial discretisation of this problem by using a Chebyshev spectral method, so that the resulting discreet problem is well posed, it is necessary that the discreet spaces of velocity and pressure verify a compatibility condition inf-sup of Brezzi [

To answer this question of compatibility condition, we use the spectral method

In this section we present the basic principle of the method

So for a given positive integers N and M we denote by

The

In this paper, we consider Chebyshev polynomials and choose the Chebyshev-Gauss-Lobatto mesh defined by:

Then, we consider the velocity at

Let us making the following space approximation for

We approach the first and secondary operators of derivation of

where

order 1

where

Let us consider the following approximation spaces:

where

We define by:

Then the prediction stage (9)-(10) decomposes itself in two-Helmholtz problems for each components of the predicted velocity with Dirichlet boundary conditions:

The Chebyshev collocation approximation of Helmholtz problems (15) and (17) is given by:

and

Multiplying these equations by

and

where

Let us denote by:

Then, we can rewrite Equations (21) and (22) by:

and

Systems (23) and (24) are solving by using diagonalisation method [

Let us denote by

and

where

Multiplying the Equation (23) on the left by

we deduce that:

Let us denote by

From (25) and (26), we deduce:

and multiplying this equation on the right by

so that, we deduce the following equation:

Denoting by

using relation (26), we obtain:

Then, we deduce:

We compute completely

1) Compute

2) Compute

3) Compute

4) Compute

5) Compute

When applying the same algorithm to Equation (24), we can compute completely

In order to make the projection stage, we define:

then we can rewrite Equations (12)-(13) by:

with boundary conditions:

So, while noting:

then by using spectral method

where

with boundary conditions :

Let us denote by:

Then, we obtain the following matrix formulation for Equations (39)-(41), given by:

where

Reformulating Equations (42), (43) and (44), we deduce :

where

We solve Equation (47) by using the same strategy using for solving Equation (21) and (22). Then we deter- mine completely the P matrix for the pressure and deduce the matrixes W and V containing the values of the first and the second components of velocity, respectively from Equations (45) and (46).

Let us denote by

We denote by

We can rewrite Equation (48) by:

where:

where

And while denoting by

To make the approximation of Equations (6)-(8), we suppose that the strong domain occupied by sand dune is

parameterized by

What brings us to consider another grid to approach the dune height by using new grid

Let us denote by

Denoting by

we obtain the following matrix formulation:

where:

and while denoting by :

For the numerical simulation, we consider an experimental solution on the one hand for the Navier-Stokes equations and other for the mass density and the dune height.

For example:

We take

the calculated fields, we give the evolution of the temporal error

integration in time of this error is initialized while taking the fields to the instants

We represent temporal errors according to the first components

components

We also represent the temporal errors for the mass density of sand grains

The profile of the dune height is represented at

We have solved numerically a mathematical model of sand dune formation in a surface water to incompressible out-flows in two space dimensions. This model couples the Navier-Stokes equations governing the incompressi- ble out-flows in two-dimension of space and the Prigozhin equation that describes the evolution of a sand dune in a surface water. One of the difficulties of this approach resides in the treatment of the pressure which appears only in Navier-Stokes equations as Lagrange multiplier. We used a Chebyshev projection scheme following a spectral approach

In our future works, we count to pass in dimension 3 of space and to put a optimal control in place to deter- mine the optimal height of sand dune in a surface water, from which other dunes can be formed in the fluid.