An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation (Communications in Computational Physics, 2020)
This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of theCahn-Hilliard model needs very long time to reach the steady state, and therefore largetime-stepping methods become useful. The main objective of this work is to constructthe unconditionally energy stable finite difference scheme so that the large time stepscan be used in the numerical simulations. The equation is discretized by the centraldifference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative.An error estimate for the numerical solution is also obtained with second order inboth space and time. By using this energy stable scheme, an adaptive time-steppingstrategy is proposed, which selects time steps adaptively based on the variation of thefree energy against time. The numerical experiments are presented to demonstrate theeffectiveness of the adaptive time-stepping approach.