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

TL;DR

This paper proves the energy stability and phase separation of a semi-implicit numerical scheme for the 2D Cahn-Hilliard equation with logarithmic potentials, a first rigorous result for such singular potentials.

Contribution

It provides the first rigorous proof of stability and phase separation for a semi-implicit scheme applied to the Cahn-Hilliard equation with logarithmic potentials.

Findings

Proves strict phase separation under certain time step constraints.

Establishes energy stability of the semi-implicit scheme.

First rigorous analysis for semi-implicit discretization with singular potentials.

Abstract

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