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

作　　者：Chaoyu Quan Tao Tang Boyi Wang Jiang Yang 

机构地区：[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 518055,Guangdong,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

出　　处：《Communications in Computational Physics》2023年第4期962-991,共30页计算物理通讯（英文）

基　　金：partially supported by the National Natural Science Foundation of China/Hong Kong RGC Joint Research Scheme(NSFC/RGC 11961160718);the fund of the Guangdong Provincial Key Laboratory of Computational Science And Material Design(No.2019B030301001);supported in part by the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science under UIC 2022B1212010006;supported by the National Science Foundation of China(NSFC)Grant No.12271240;supported by NSFC Grant 12271241;Guangdong Basic and Applied Basic Research Foundation(No.2023B1515020030);Shenzhen Science and Technology Program(Grant No.RCYX20210609104358076).

摘　　要：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.

关 键 词：Time-fractional Allen-Cahn equation energy dissipation L1 approximation L2 approximation 

分 类 号：O17[理学—数学]