Journal
SIAM JOURNAL ON SCIENTIFIC COMPUTING
Volume 41, Issue 6, Pages A3757-A3778Publisher
SIAM PUBLICATIONS
DOI: 10.1137/18M1203560
Keywords
time-fractional phase-field equations; Allen-Cahn equation; Cahn-Hilliard equation; MBE model; energy dissipation law; maximum principle
Categories
Funding
- NNSF of China [11688101, 11771439, 91530322, 91630312, 91630203, 11571351, 11731006]
- China National Program on Key Basic Research Project [2015CB856003]
- Science Challenge Project [TZ2018001]
- NCMIS
- Youth Innovation Promotion Association (CAS)
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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 -alpha/3, where alpha is the fractional parameter.
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