On energy dissipation theory and numerical stability for time-fractional phase field equations

TL;DR

This paper establishes an energy dissipation law for time-fractional phase field models at both continuous and discrete levels and proposes stable finite difference schemes, supported by numerical validation and coarsening rate analysis.

Contribution

It proves the energy dissipation law for time-fractional phase field models and develops energy-stable finite difference schemes, addressing a key open problem in the field.

Findings

Energy dissipation law is valid for time-fractional models.

Finite difference schemes inherit energy stability.

Coarsening rate follows a -α/3 power law.

Abstract

For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall 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 gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. 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 both time-fractional Cahn-Hilliard equation and time-fractional molecular beam epitaxy model, while there exists a $-\alpha/3$ power law coarsening stage.