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

Why large time-stepping methods for the Cahn-Hilliard equation is stable