A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations

Quan，Chaoyu1; Tang，Tao2,3; Wang，Boyi4,5; Yang，Jiang4,6

2023-04-01

10.4208/cicp.OA-2022-0148

Communications in Computational Physics   Impact Factor & Quartile

1815-2406

1991-7120

In this article, we study the energy dissipation property of time-fractional Allen-Cahn equation. On the continuous level, we propose an upper bound of energy that decreases with respect to time and coincides with the original energy at t=0 and as t tends to ∞. This upper bound can also be viewed as a nonlocal-in-time modified energy which is the summation of the original energy and an accumulation term due to the memory effect of time-fractional derivative. In particular, the decrease of the modified energy indicates that the original energy indeed decays w.r.t. time in a small neighborhood at t=0. We illustrate the theory mainly with the time-fractional Allen- Cahn equation but it could also be applied to other time-fractional phase-field models such as the Cahn-Hilliard equation. On the discrete level, the decreasing upper bound of energy is useful for proving energy dissipation of numerical schemes. First-order L1 and second-order L2 schemes for the time-fractional Allen-Cahn equation have similar decreasing modified energies, so that stability can be established. Some numerical results are provided to illustrate the behavior of this modified energy and to verify our theoretical results.

energy dissipation L1 approximation L2 approximation Time-fractional Allen-Cahn equation

SCI

English

First ; Corresponding

National Natural Science Foundation of China/Hong Kong RGC Joint Research Scheme[NSFC/RGC 11961160718] ; Guangdong Provincial Key Laboratory of Computational Science And Material Design[2019B030301001] ; National Science Foundation of China (NSFC)[12271241] ; NSFC[2023B1515020030] ; Guangdong Basic and Applied Basic Research Foundation[RCYX20210609104358076] ; Shenzhen Science and Technology Program[UIC 2022B1212010006] ; null[12271240]

Physics

Physics, Mathematical

WOS:000993886300002

GLOBAL SCIENCE PRESS

2-s2.0-85161279707

Scopus

1.International Center for Mathematics,Southern University of Science and Technology,Shenzhen,518055,China
2.Division of Science and Technology,BNU-HKBU United International College,Zhuhai,519087,China
3.Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science,BNU-HKBU United International College,Zhuhai,519087,China
4.Department of Mathematics,Southern University of Science and Technology,Shenzhen,Guangdong,518055,China
5.Department of Mathematics,National University of Singapore,Singapore,119076,Singapore
6.Guangdong Provincial Key Laboratory of Computational Science and Material Design,Southern University of Science and Technology,Shenzhen,518055,China

Quan，Chaoyu,Tang，Tao,Wang，Boyi,et al. A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations[J]. Communications in Computational Physics,2023,33(4):962-991.

Quan，Chaoyu,Tang，Tao,Wang，Boyi,&Yang，Jiang.(2023).A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations.Communications in Computational Physics,33(4),962-991.

Quan，Chaoyu,et al."A Decreasing Upper Bound of the Energy for Time-Fractional Phase-Field Equations".Communications in Computational Physics 33.4(2023):962-991.