I know that stability and convergence are 2 faces of the same coin. however when it comes to convergence, I can't find why it is important for it to be monotonous. Maximum always decrease and minimum always increase. Is it because it will sensitive to perturbations? or will the stability of the solution depend on the initial conditions. If so, why does instability happen if the criterion of stability (Fourier's number) is satisfied?
I'm a bit confused, since this something i'm dealing with, and I can't understand it intuitively.
Thank you!