Now, the v velocity is not a constant and the 2-D Burgers equation possesses
two nolinear terms. We propose here to calculate this speed with the continuity
equation defined by

(2.4)

So the new problem to solve has no physical representation. In fact,
if we consider a physical problem describes by the 2-D Burgers equation
issue to the simplification of the Navier-Stokes equations, it signified
that the v velocity does not depends of x, y and z variables. The introduction
of the continuity equation imposed the dependence of these variables for
v velocity. Physically, it is impossible to consider the continuity equation
and the Burgers' equation together.
However, it may be interresting to take a non constant v velocity in
order to see a not uniform transversal convection.

We can see that the v velocity is all more so great since the u velocity
gradients are important. This is true where the shock is because it is
defined by its discontinuity, so its velocity gradients. With this v velocity
we can expect to see the highest transversal convection where the shock
is.

Numerically, we approximate the term
by a central difference defined by

And v can be approximated by a trapezoidal method defined by

To solve the problem, we must initialized the calculation (see appendix
Appendix
B) by an arbitrary v velocity : we choose.