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We consider the two-dimensional Cahn-Hilliard equation with logarithmicpotentials and periodic boundary conditions. We employ the standardsemi-implicit numerical scheme which treats  the linear fourth-order dissipation term implicitly and the nonlinear termexplicitly. Under natural  constraints on the time step we prove strict phase separation and energystability of the semi-implicit scheme. This appears to be the first rigorousresult for the semi-implicit discretization  of the Cahn-Hilliard equation with singular potentials.

Stability of the semi-implicit method for the Cahn-Hilliard equation with logarithmic potentials