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

Abstract:   For the time-fractional phase field models,  the corresponding energy dissipation law has not been well studied on both the continuous level and the discrete level.  In this talk, we shall address these open issues. More precisely, we prove 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 gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models.  Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter $\alpha$ reveals that there are several coarsening stages for all the three models, while there exist a $-\alpha/3$ power law coarsening stage for both the time-fractional Cahn-Hilliard equation and the time-fractional molecular beam epitaxy model.

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