Computationally efficient adaptive time step method for the Cahn-Hilliard equation

In this work, we propose a fast and efficient adaptive time step procedure for the Cahn-Hilliard equation. The temporal evolution of the Cahn-Hilliard equation has multiple time scales. For spinodal decomposition simulation, an initial random perturbation evolves on a fast time scale, and later coarsening evolves on a very slow time scale. Therefore, if a small time step is used to capture the fast dynamics, the computation is quite costly. On the other hand, if a large time step is used, fast time evolutions may be missed. Hence, it is essential to use an adaptive time step method to simulate phenomena with multiple time scales. The proposed time adaptivity algorithm is based on the discrete maximum norm of the difference between two consecutive time step numerical solutions. Numerical experiments in one, two, and three dimensions are presented to demonstrate the performance and effectiveness of the adaptive time-stepping algorithm.