Unconditionally strong energy stable scheme for Cahn-Hilliard equation with second-order temporal accuracy

Abstract

We propose an unconditionally gradient stable scheme for solving the Cahn-Hilliard equation with second-order accuracy in time using the effective time-step analysis. The conventional convex splitting scheme is one of the most well-known methods for solving gradient flows, and it guarantees both energy stability and first-order accuracy in time. Recently, there are some researches, which are extended to the second-order accuracy; however, most of results provide the proof of energy stability for a modified (pseudo) energy or in a weak sense. In this paper, we prove the energy stability of the proposed method with respect to the Ginzburg-Landau free energy functional. Moreover, the unique solvability, mass conservation, and accuracy of the proposed scheme are also proven based on the convex splitting approach. The numerical experiments are presented to show convergence rate, mass conservation, energy stability, and phase separation are in good agreement with the theory.