The convex-concave splitting scheme for the Allen-Cahn equation moves interfaces only on a time scale of order epsilon squared, so its guaranteed stability comes from freezing the interface motion.
On the slowness of phase boundary motion in one space dimension
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Convex-concave splitting for the Allen-Cahn equation leads to $\varepsilon^2$-slow movement of interfaces
The convex-concave splitting scheme for the Allen-Cahn equation moves interfaces only on a time scale of order epsilon squared, so its guaranteed stability comes from freezing the interface motion.