Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations

Hou, Tianliang; Tang, Tao; Yang, Jiang

doi:10.1007/s10915-017-0396-9

Cited by 141 publications

(79 citation statements)

References 21 publications

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“…Several numerical techniques have been recently developed for space and time non-local versions of equation (1.3), most of them based on finite differences or spectral methods [28,21,35,22,27,7]. Also, numerical methods have been studied for nonlocal versions of related phase separation models, like the Cahn-Hilliard equation [5,6].…”

Section: Introductionmentioning

confidence: 99%

Numerical approximations for a fully fractional Allen–Cahn equation

Acosta

Bersetche

2021

ESAIM: M2AN

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A finite element scheme for an entirely fractional Allen-Cahn equation is introduced and analyzed. In the proposed nonlocal model, the Caputo fractional in-time derivative and the fractional Laplacian replace the standard local operators. Piecewise linear finite elements and convolution quadratures are the basic tools involved in the presented numerical method. Error analysis and implementation issues are addressed together with the needed results of regularity for the continuous model. Also, the asymptotic behavior of solutions, for a vanishing diffusion parameter, is analyzed within the framework of the Γ-convergence theory.On the other hand, even though (1.1) makes perfect sense for n = 1, lateral versions are necessary for dealing with the time variable. In such a case, the so-called 2010 Mathematics Subject Classification. 65R20, 65M60, 35R11.

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Section: Introductionmentioning

confidence: 99%

Numerical approximations for a fully fractional Allen–Cahn equation

Acosta

Bersetche

2021

ESAIM: M2AN

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“…They concluded that the alternative model could provide more accurate description of anomalous diffusion processes and sharper interfaces than the classical model. Hou et al [13] showed that a fractional in space Allen-Cahn equation could be viewed an L 2 gradient flow for the fractional analogue version of Ginzburg-Landau free energy function. They proved the energy decay property and the maximum principle of continuous problem.…”

Section: Introductionmentioning

confidence: 99%

Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation

Liao

Zhang

2020

Adv Comput Math

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Two fast L1 time-stepping methods, including the backward Euler and stabilized semi-implicit schemes, are suggested for the time-fractional Allen-Cahn equation with Caputo's derivative. The time mesh is refined near the initial time to resolve the intrinsically initial singularity of solution, and unequal time-steps are always incorporated into our approaches so that an adaptive time-stepping strategy can be used in long-time simulations. It is shown that the proposed schemes using the fast L1 formula preserve the discrete maximum principle. Sharp error estimates reflecting the time regularity of solution are established by applying the discrete fractional Grönwall inequality and global consistency analysis. Numerical experiments are presented to show the effectiveness of our methods and to confirm our analysis.

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“…Shen et al [33] generalized the results presented in [37] to the case of the Allen-Cahn-like equation in a more abstract form with the potential and mobility satisfying some certain conditions. Hou et al [26] studied the numerical approximation of the fractional Allen-Cahn equation by considering the conventional Crank-Nicolson scheme. They proved that the Crank-Nicolson scheme preserves the maximum principle and this is the first work on the second order schemes preserving the maximum principle.…”

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confidence: 99%

Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen--Cahn Equation

Xiao

et al. 2019

SIAM J. Numer. Anal.

219

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The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete maximum principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadraturebased finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.

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Numerical Analysis of Fully Discretized Crank–Nicolson Scheme for Fractional-in-Space Allen–Cahn Equations

Cited by 141 publications

References 21 publications

Numerical approximations for a fully fractional Allen–Cahn equation

Numerical approximations for a fully fractional Allen–Cahn equation

Simple maximum principle preserving time-stepping methods for time-fractional Allen-Cahn equation

Maximum Principle Preserving Exponential Time Differencing Schemes for the Nonlocal Allen--Cahn Equation

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