Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods

In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order α ∈ ( 0 , 1 ) . First, the well-posedness and (limited) smoothing property are studied, by using the maximal L p regularity of...

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Vydáno v:	Journal of scientific computing Ročník 85; číslo 2; s. 42
Hlavní autoři:	Du, Qiang, Yang, Jiang, Zhou, Zhi
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.11.2020 Springer Nature B.V
Témata:	Algorithms Computational Mathematics and Numerical Analysis Energy dissipation Estimates Free energy Gradient flow Inequality Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Maximum principle Molecular beam epitaxy Numerical analysis Numerical methods Quadratures Regularity Splitting Theoretical 65R20 Energy dissipation 65M70 Error estimate Time-fractional Allen–Cahn Regularity Time stepping scheme
ISSN:	0885-7474, 1573-7691
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Abstract	In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order α ∈ ( 0 , 1 ) . First, the well-posedness and (limited) smoothing property are studied, by using the maximal L p regularity of fractional evolution equations and the fractional Grönwall’s inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall’s inequality and maximal ℓ p regularity, we prove that the convergence rates of those time-stepping schemes are O ( τ α ) without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen–Cahn dynamics.
AbstractList	In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order α∈(0,1). First, the well-posedness and (limited) smoothing property are studied, by using the maximal Lp regularity of fractional evolution equations and the fractional Grönwall’s inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall’s inequality and maximal ℓp regularity, we prove that the convergence rates of those time-stepping schemes are O(τα) without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen–Cahn dynamics. In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional derivative with order α ∈ ( 0 , 1 ) . First, the well-posedness and (limited) smoothing property are studied, by using the maximal L p regularity of fractional evolution equations and the fractional Grönwall’s inequality. We also show the maximum principle like their conventional local-in-time counterpart, that is, the time-fractional equation preserves the property that the solution only takes value between the wells of the double-well potential when the initial data does the same. Second, after discretizing the fractional derivative by backward Euler convolution quadrature, we develop several unconditionally solvable and stable time stepping schemes, such as a convex splitting scheme, a weighted convex splitting scheme and a linear weighted stabilized scheme. Meanwhile, we study the discrete energy dissipation property (in a weighted average sense), which is important for gradient flow type models, for the two weighted schemes. In addition, we prove the fractional energy dissipation law for the gradient flow associated with a convex free energy. Finally, using a discrete version of fractional Grönwall’s inequality and maximal ℓ p regularity, we prove that the convergence rates of those time-stepping schemes are O ( τ α ) without any extra regularity assumption on the solution. We also present extensive numerical results to support our theoretical findings and to offer new insight on the time-fractional Allen–Cahn dynamics.
ArticleNumber	42
Author	Zhou, Zhi Yang, Jiang Du, Qiang
Author_xml	– sequence: 1 givenname: Qiang surname: Du fullname: Du, Qiang organization: Department of Applied Physics and Applied Mathematics, Columbia University – sequence: 2 givenname: Jiang orcidid: 0000-0002-6431-7483 surname: Yang fullname: Yang, Jiang email: yangj7@sustech.edu.cn organization: Department of Mathematics, SUSTech International Center for Mathematics, Southern University of Science and Technology, Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology – sequence: 3 givenname: Zhi surname: Zhou fullname: Zhou, Zhi organization: Department of Applied Mathematics, The Hong Kong Polytechnic University
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Snippet	In this work, we consider a time-fractional Allen–Cahn equation, where the conventional first order time derivative is replaced by a Caputo fractional...
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SubjectTerms	Algorithms Computational Mathematics and Numerical Analysis Energy dissipation Estimates Free energy Gradient flow Inequality Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Maximum principle Molecular beam epitaxy Numerical analysis Numerical methods Quadratures Regularity Splitting Theoretical
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Title	Time-Fractional Allen–Cahn Equations: Analysis and Numerical Methods
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