Time-fractional Allen-Cahn and Cahn-Hilliard phase-field models and their numerical investigation

We study (time) fractional Allen-Cahn and Cahn-Hilliard phase-field models to account for the anomalously subdiffusive transport behavior in heterogeneous porous materials or memory effect of certain materials. We develop an efficient finite difference scheme and a Fourier spectral scheme to effectively treat the significantly increased memory requirement and computational complexity, which arise due to the nonlocal behavior of the time-fractional models. For time fractional Cahn-Hilliard model, we observe from the numerical results that the bigger the fractional order a is, the faster the energy decays. However, for time fractional Allen-Cahn model, we derived an opposite conclusion. Moreover, we also study the coarsening dynamics for time fractional Cahn-Hilliard model, numerical results reveal that the scaling law for the energy decays as O(t(-1/3)), which is consistent with the well-known result O(t(-1/3)) for integer order Cahn-Hilliard model. (C) 2018 Elsevier Ltd. All rights reserved.