Comparison study of numerical methods for solving the Allen–Cahn equation

Jeong, Darae; Lee, Seunggyu; Lee, Dongsun; Shin, Jeeyoung; Kim, Junseok

doi:10.1016/j.commatsci.2015.09.005

Cited by 34 publications

(18 citation statements)

References 27 publications

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“…CH equation is thus written as a set of two coupled equations making the system complex and increasing the degrees of freedom. A comparison between the AC and CH equations has been carried out in [35,36]. A review of the phase-field models for multiphase flows can be found in [37].…”

Section: Introductionmentioning

confidence: 99%

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

Joshi

Jaiman

2018

Journal of Computational Physics

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We present a positivity preserving variational scheme for the phase-field modeling of incompressible two-phase flows with high density ratio and using meshes of arbitrary topology. The variational finite element technique relies on the Allen-Cahn phase-field equation for capturing the phase interface on a fixed mesh with a mass conservative and energy-stable discretization. Mass is conserved by enforcing a Lagrange multiplier which has both temporal and spatial dependence on the solution of the phase-field equation. The spatial part of the Lagrange multiplier is written as a mid-point approximation to make the scheme energy-stable. This enables us to form a conservative, energy-stable and positivity preserving scheme. The proposed variational technique reduces spurious and unphysical oscillations in the solution while maintaining second-order spatial accuracy. To model a generic two-phase free-surface flow, we couple the Allen-Cahn phase-field equation with the Navier-Stokes equations. Comparison of results between standard linear stabilized finite element method and the present variational formulation shows a remarkable reduction of oscillations in the solution while retaining the boundedness of the phase-indicator field. We perform a standalone test to verify the accuracy and stability of the Allen-Cahn two-phase solver. Standard two-phase flow benchmarks such as Laplace-Young law and sloshing tank problem are carried out to assess the convergence and accuracy of the coupled Navier-Stokes and Allen-Cahn solver. Two-and three-dimensional dam break problem are then solved to assess the scheme for the problem with topological changes of the air-water interface on unstructured meshes. Finally, we demonstrate the phase-field solver for a practical problem of wavestructure interaction in offshore engineering using general three-dimensional unstructured meshes.

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Section: Introductionmentioning

confidence: 99%

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

Joshi

Jaiman

2018

Journal of Computational Physics

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“…In this work, we introduce new schemes, based on a splitting (also referred to as splitting-up) strategy, see for instance [5,6,24,25] in the SPDE case, and [43,51] for the deterministic Allen-Cahn equation. Indeed, the solution of the ordinary differential equationż = z − z 3 has a known explicit expression.…”

Section: Charles-edouard Bréhier and Ludovic Goudenègementioning

confidence: 99%

“…In the case of Allen-Cahn equations, our strategy, detailed below, consists in replacing these implicit steps with the exact solution of the flow associated with Ψ 0 , in the spirit of [43,51].…”

Section: Wiener Processmentioning

confidence: 99%

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

Bréhier¹,

Goudenège²

2019

Discrete &Amp; Continuous Dynamical Systems - B

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We introduce and analyze an explicit time discretization scheme for the one-dimensional stochastic Allen-Cahn, driven by space-time white noise. The scheme is based on a splitting strategy, and uses the exact solution for the nonlinear term contribution. We first prove boundedness of moments of the numerical solution. We then prove strong convergence results: first, L 2 (Ω)-convergence of order almost 1/4, localized on an event of arbitrarily large probability, then convergence in probability of order almost 1/4. The theoretical analysis is supported by numerical experiments, concerning strong and weak orders of convergence.

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“…Numerical Solution. We consider the Fourier spectral method in space and the linear convex splitting scheme, which is known as stable and uniquely solvable one [17], in time. First, the temporal discretization of (8) is written as follows:…”

Section: Modification To Image Segmentationmentioning

confidence: 99%

Image Segmentation Based on Modified Fractional Allen–Cahn Equation

Lee

2019

Mathematical Problems in Engineering

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We present the image segmentation model using the modified Allen–Cahn equation with a fractional Laplacian. The motion of the interface for the classical Allen–Cahn equation is known as the mean curvature flows, whereas its dynamics is changed to the macroscopic limit of Lévy process by replacing the Laplacian operator with the fractional one. To numerical implementation, we prove the unconditionally unique solvability and energy stability of the numerical scheme for the proposed model. The effect of a fractional Laplacian operator in our own and in the Allen–Cahn equation is checked by numerical simulations. Finally, we give some image segmentation results with different fractional order, including the standard Laplacian operator.

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Comparison study of numerical methods for solving the Allen–Cahn equation

Cited by 34 publications

References 27 publications

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

A positivity preserving and conservative variational scheme for phase-field modeling of two-phase flows

Analysis of some splitting schemes for the stochastic Allen-Cahn equation

Image Segmentation Based on Modified Fractional Allen–Cahn Equation

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