The previous study shows that it doesn't necessary to solve equation
with a Reynolds number equal to 100 because the shock never appears. So,
we take as the previous section Re=1000. Results concerning the resolution
of the 2D-Burgers equation coupled with the continuity equation were reported
on graphics located in the section "Evolution
of u velocity". The difference with a constant v velocity is not striking,
but we can see the steepening of the wave front as the previous case. As
we can expect, those results show that the transversal propagation (in
y direction) carries out only where the shock is located. In fact, (see
the expression of v velocity eq. 2.5)
the y propagation doesn't exist where the u velocity gradients are non
exixtant in particular on the ``bottom'' and on the ``top'' of the sinus.
Likewise, we can imagine that the propagation in y direction is negative
where the u velocity gradients were positive (on the x boundary conditions
for example). And this propagation is positive where the u velocity gradients
are negative and all the more so great since the u velocity gradients are
important. To visualize those effects we can just see the evolution of
v velocity in the section "Evolution of v
velocity".

But those phenomenons are not very well visible on the previous graphics, it is most appropriate to display the corresponding animations :

Click here to see animations of the steepening and the propagation of the wave front with Re=1000 for u velocity and for v velocity.