  We consider the long time behavior of the semidiscrete scheme for the
Perona-Malik equation in dimension one. We prove that approximated solutions
converge, in a slow time scale, to solutions of a limit problem. This limit
problem evolves piecewise constant functions by moving their plateaus in the
vertical direction according to a system of ordinary differential equations.
  Our convergence result is global-in-time, and this forces us to face the
collision of plateaus when the system singularizes.
  The proof is based on energy estimates and gradient-flow techniques,
according to the general idea that "the limit of the gradient-flows is the
gradient-flow of the limit functional". Our main innovations are a uniform
H\"{o}lder estimate up to the first collision time included, a well preparation
result with a careful analysis of what happens at discrete level during
collisions, and renormalizing the functionals after each collision in order to
have a nontrivial Gamma-limit for all times.
