  We consider the Cahn-Hilliard equation with standard double-well potential.
We employ a prototypical class of first order in time semi-implicit methods
with implicit treatment of the linear dissipation term and explicit
extrapolation of the nonlinear term. When the dissipation coefficient is held
small, a conventional wisdom is to add a judiciously chosen stabilization term
in order to afford relatively large time stepping and speed up the simulation.
In practical numerical implementations it has been long observed that the
resulting system exhibits remarkable stability properties in the regime where
the stabilization parameter is $\mathcal O(1)$, the dissipation coefficient is
vanishingly small and the size of the time step is moderately large. In this
work we develop a new stability theory to address this perplexing phenomenon.
