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The physical problem

We solve 2-D Burgers equation in a square field integration limited by $\Omega = (x,y)\in [(0,1) \times (0,1)]$. The u velocity depends of time because we consider an unsteady problem with two space variables : u=u(x,y,t)

$\bullet$ 2-D Burgers equation :
 

\begin{displaymath}\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial......tial x^{2}} +\frac{{\partial}^{2} u}{\partial y^{2}} \right)\end{displaymath} (2.1)
 with $v$ the transversal velocity : $v(x,y,t)$

In order to trigger the shock, we take an initial condition sinus type in the x direction. We choose to deaden this sinus by an exponential term in the y direction. Thus, we will be able to see the propagation of the wave front in this direction :

$\bullet$ Initial condition :
 

\begin{displaymath}u(x,y,0)=sin(2 \pi x) e^{-ky} \ \ , \ k>0 \ \ \ \forall \ (x,y) \ \in \ [(0,1)\times(0,1)]\end{displaymath}

(2.2)
 In order to pose well the problem, we must define some boundary conditions. Those are four because this is a second order problem with two space dimensions.

The boundary conditions in x=0 and x=1 are fixed at zero, and these  in y=0 always corresponds to a sinus in order to cause the steepening of the wave front. On the other hand, to be compatible with the initial condition, the boundary condition in y=1 follows an exponential decreasing law of the sinus :

$\bullet$ Boundary conditions : 


 
\begin{displaymath}\forall \, t \, \in \, [0,T] \ \left\{\begin{array}{ccll}...... x) e^{-k} & \forall \, x \, \in \, [0,1]\end{array} \right.\end{displaymath} (2.3)



Initial condition on u velocity

(We can see this initial condition in his original size in clicking here)

The problem thus proposed is however not completely well posed because the advection v velocity is not defined. Precisely, one proposes in this study to test several cases for this transversal velocity.
 


Subsections
nextuppreviouscontents
Next:Resolution with a constant Up:The 2-D Burgers Equation Previous:IntroductionContents
 
Alban Depoutre
2000-11-21