A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations

Song, Fangying; Xu, Chuanju; Karniadakis, George Em

doi:10.1016/j.cma.2016.03.018

Cited by 96 publications

(46 citation statements)

References 51 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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“…Such an energy dissipation low plays an important role in developing stable numerical methods for dissipation systems due to its importance to the long time numerical simulation, see e.g., [12,13,14,44,36,49,16,46,42] and references therein. In recent years, fractional-type phase-field models have attracted more and more attentions [1,2,3,7,30,38,23]. For instance, consider a fractional type free energy [38] E α (φ) := Ω ε 2 2 |∇ α φ| 2 + F (φ) dx, (1.5) where ∇ α is the fractional gradient ∇ α = ( ∂ α ∂x1 , ..., ∂ α ∂x d ) and { ∂ α ∂x k } k are fractional derivatives.…”

mentioning

confidence: 99%

“…In recent years, fractional-type phase-field models have attracted more and more attentions [1,2,3,7,30,38,23]. For instance, consider a fractional type free energy [38] E α (φ) := Ω ε 2 2 |∇ α φ| 2 + F (φ) dx, (1.5) where ∇ α is the fractional gradient ∇ α = ( ∂ α ∂x1 , ..., ∂ α ∂x d ) and { ∂ α ∂x k } k are fractional derivatives. One is interested in the following space-fractional CH equation…”

mentioning

confidence: 99%

On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

Tang¹,

Yu²,

Zhou³

2019

SIAM J. Sci. Comput.

126

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For the time-fractional phase field models, the corresponding energy dissipation law has not been settled on both the continuous level and the discrete level. In this work, we shall address this open issue. More precisely, we prove for the first time that the time-fractional phase field models indeed admit an energy dissipation law of an integral type. In the discrete level, we propose a class of finite difference schemes that can inherit the theoretical energy stability. Our discussion covers the time-fractional gradient systems, including the time-fractional Allen-Cahn equation, the time-fractional Cahn-Hilliard equation, and the time-fractional molecular beam epitaxy models. Numerical examples are presented to confirm the theoretical results. Moreover, a numerical study of the coarsening rate of random initial states depending on the fractional parameter α reveals that there are several coarsening stages for both time-fractional Cahn-Hilliard equation and timefractional molecular beam epitaxy model, while there exists a −α/3 power law coarsening stage.

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“…For spatially nonlocal phase field models, due to the spread of nonlocal interactions, the phase field variables may no longer be as smooth as their local counterpart. They may also lead to narrower interfacial region and permit singularities across the interface or at the defects [159,239,394]. 7.5.2.…”

Section: Fluid and Solid Mechanicsmentioning

confidence: 99%

The phase field method for geometric moving interfaces and their numerical approximations

Feng

2020

Geometric Partial Differential Equations - Part I

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This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science. Key words and phrases. Phase field method, geometric law, curvature-driven flow, geometric nonlinear PDEs, finite difference methods, finite element methods, spectral methods, discontinuous Galerkin methods, adaptivity, coarse and fine error estimates, convergence of numerical interfaces, nonlocal and stochastic phase field models, microstructure evolution, biology and image science applications.

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A fractional phase-field model for two-phase flows with tunable sharpness: Algorithms and simulations

Cited by 96 publications

References 51 publications

Numerical approximations for a fully fractional Allen–Cahn equation

Numerical approximations for a fully fractional Allen–Cahn equation

On Energy Dissipation Theory and Numerical Stability for Time-Fractional Phase-Field Equations

The phase field method for geometric moving interfaces and their numerical approximations

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