2-D Burgers equation :

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 :

Initial condition :

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 :

Boundary conditions :

(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.