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 AN ENERGY STABLE AND MAXIMUM BOUND PRESERVING SCHEME WITH VARIABLE TIME STEPS FOR TIME FRACTIONAL ALLEN--CAHN EQUATION Liao, Hong-lin1,2; Tang, Tao3,4,5; Zhou, Tao6 2021 Source Publication SIAM JOURNAL ON SCIENTIFIC COMPUTING ISSN 1064-8275 Volume 43Issue:5Pages:A3503-A3526 Abstract In this work, we propose a Crank-Nicolson-type scheme with variable steps for the time fractional Allen--Cahn equation. The proposed scheme is shown to be unconditionally stable (in a variational energy sense) and is maximum bound preserving. Interestingly, the discrete energy stability result obtained in this paper can recover the classical energy dissipation law when the fractional order \alpha -1. That is, our scheme can asymptotically preserve the energy dissipation law in the \alpha -1 limit. This seems to be the first work on a variable time-stepping scheme that can preserve both the energy stability and the maximum bound principle. Our Crank-Nicolson scheme is built upon a reformulated problem associated with the Riemann-Liouville derivative. As a byproduct, we build up a reversible transformation between the L1-type formula of the RiemannLiouville derivative and a new L1-type formula of the Caputo derivative with the help of a class of discrete orthogonal convolution kernels. This is the first time such a discrete transformation is established between two discrete fractional derivatives. We finally present several numerical examples with an adaptive time-stepping strategy to show the effectiveness of the proposed scheme. Keyword time-fractional Allen--Cahn equation asymptotic preserving energy stability adaptive time stepping max-imum principle DOI 10.1137/20M1384105 Indexed By SCI Language 英语 Funding Project NSF of China[11822111] ; NSF of China[11688101] ; NSF of China[12071216] ; NSF of China[11731006] ; NSF of China[K20911001] ; Youth Innovation Promotion Association of the CAS ; science challenge project[TZ2018001] WOS Research Area Mathematics WOS Subject Mathematics, Applied WOS ID WOS:000712863700024 Publisher SIAM PUBLICATIONS Citation statistics Document Type 期刊论文 Identifier http://ir.amss.ac.cn/handle/2S8OKBNM/59514 Collection 中国科学院数学与系统科学研究院 Corresponding Author Zhou, Tao Affiliation 1.Nanjing Univ Aeronaut & Astronaut, Dept Math, Nanjing 211106, Peoples R China2.MIIT, Key Lab Math Modelling & High Performance Comp Ai, Nanjing 211106, Peoples R China3.BNU HKBU United Int Coll, Div Sci & Technol, Zhuhai, Guangdong, Peoples R China4.Southern Univ Sci & Technol, Dept Math, Shenzhen, Guangdong, Peoples R China5.Southern Univ Sci & Technol, Int Ctr Math, Shenzhen, Guangdong, Peoples R China6.Chinese Acad Sci, Acad Math & Syst Sci, LSEC, Inst Computat Math & Sci Engn Comp, Beijing 100190, Peoples R China Recommended CitationGB/T 7714 Liao, Hong-lin,Tang, Tao,Zhou, Tao. AN ENERGY STABLE AND MAXIMUM BOUND PRESERVING SCHEME WITH VARIABLE TIME STEPS FOR TIME FRACTIONAL ALLEN--CAHN EQUATION[J]. SIAM JOURNAL ON SCIENTIFIC COMPUTING,2021,43(5):A3503-A3526. APA Liao, Hong-lin,Tang, Tao,&Zhou, Tao.(2021).AN ENERGY STABLE AND MAXIMUM BOUND PRESERVING SCHEME WITH VARIABLE TIME STEPS FOR TIME FRACTIONAL ALLEN--CAHN EQUATION.SIAM JOURNAL ON SCIENTIFIC COMPUTING,43(5),A3503-A3526. MLA Liao, Hong-lin,et al."AN ENERGY STABLE AND MAXIMUM BOUND PRESERVING SCHEME WITH VARIABLE TIME STEPS FOR TIME FRACTIONAL ALLEN--CAHN EQUATION".SIAM JOURNAL ON SCIENTIFIC COMPUTING 43.5(2021):A3503-A3526.
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