To conclude this project, I have to explain why it is of great interest for a CFD engineer. But first let's come back on the use of Newton's Method to solve non-linear systems. More precisely :
I found a link with our formation in  and , where R. Choquet consider the implicit numerical resolution of the compressible Navier-Stokes equations with the Newton's Method. We can resume what he introduced by :
We all know that this choice is not used in industrial CFD codes, but it is interesting to see that it can be a way of solving the Navier-Stokes system, even if Newton's Method can lead to fractal !...
For future session, I think that it could be interesting to study the Newton's method applied to the polynomial x->x^3+ax-1, where a is a real parameter. I know that it exists some a for which there is bifurcations, see  for more details.