On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

For the time-fractional phase-field models, the corresponding energy dissipation law has not been well studied on both the continuous and the discrete levels. In this work, we address this open issue. More precisely, we prove for the first time that the time-fractional phase-field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional Allen--Cahn equation, the time-fractional Cahn--Hilliard equation, and the time-fractional molecular beam epitaxy models. Several numerical experiments are carried out to verify the theoretical predictions. In particular, it is observed numerically that for both the time-fractional Cahn--Hilliard equation and the time-fractional molecular beam epitaxy model, there exists a coarsening stage for which the energy dissipation rate satisfies a power law scaling with an asymptotic power -α/3, where α is the fractional parameter.