A maximum-principle preserving and unconditionally energy-stable linear second-order finite difference scheme for Allen–Cahn equations

Feng, Jundong; Zhou, Yingcong; Hou, Tianliang

doi:10.1016/j.aml.2021.107179

Cited by 19 publications

(4 citation statements)

References 15 publications

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“…Since the other intrinsic property of the AC equation is the maximum-principle preserving 20 along with the energy dissipation law above, research on schemes that inherit these properties at the discrete level has been recently presented. [21][22][23][24] Feng et al developed a linear second-order scheme based on the leapfrog scheme with a stabilized term. 21 Under moderate constraints on the time step, the maximum-principle is preserved at a discrete level.…”

Section: Introductionmentioning

confidence: 99%

“…[21][22][23][24] Feng et al developed a linear second-order scheme based on the leapfrog scheme with a stabilized term. 21 Under moderate constraints on the time step, the maximum-principle is preserved at a discrete level. In Reference 22, the authors provided a high-order explicit scheme for the AC equation, which is up to the fourth-order in time and is based on the integrating factor Runge-Kutta method.…”

Section: Introductionmentioning

confidence: 99%

“…To resolve these scientific problems, several numerical solvers for the AC equation have been developed so far. Since the other intrinsic property of the AC equation is the maximum‐principle preserving 20 along with the energy dissipation law above, research on schemes that inherit these properties at the discrete level has been recently presented 21‐24 . Feng et al developed a linear second‐order scheme based on the leapfrog scheme with a stabilized term 21 .…”

Section: Introductionmentioning

confidence: 99%

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Effective time step analysis for the Allen–Cahn equation with a high‐order polynomial free energy

Lee

Yoon

Lee

et al. 2022

Numerical Meth Engineering

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An effective time step analysis to the linear convex splitting scheme for the Allen–Cahn equation with a high‐order polynomial free energy is presented in this article. Although the convex splitting scheme is unconditionally stable, using a large time step causes a time step rescaling effect, leading to delayed dynamics of the governing equation. We verify this problem by comparing it with a reformulated semi‐implicit scheme using the effective time step. Theoretical results show that the discrete energy stability and maximum‐principle hold, and the numerical results demonstrate that the time step rescaling issue can be resolved using the effective time step. We confirm that slow dynamics due to high‐order potential is alleviated by the time step modification through the results of motion by mean curvature.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Effective time step analysis for the Allen–Cahn equation with a high‐order polynomial free energy

Lee

Yoon

Lee

et al. 2022

Numerical Meth Engineering

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“…They proved that the discrete maximum principle holds under suitable mesh size and time step constraints. Feng et al [9] constructed a linear second-order finite difference scheme based on the Leap-Frog scheme. The proposed scheme is MBP-preserving and unconditionally energy-stable.…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a linear second-order finite difference scheme for space-fractional Allen–Cahn equations

et al. 2022

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In this paper, we construct a new linear second-order finite difference scheme with two parameters for space-fractional Allen–Cahn equations. We first prove that the discrete maximum principle holds under reasonable constraints on time step size and coefficient of stabilized term. Secondly, we analyze the maximum-norm error. Thirdly, we can see that the proposed scheme is unconditionally energy-stable by defining the modified energy and selecting the appropriate parameters. Finally, two numerical examples are presented to verify the theoretical results.

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Consistently and unconditionally energy-stable linear method for the diffuse-interface model of narrow volume reconstruction

Yang,

Kim

2024

Engineering with Computers

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A maximum-principle preserving and unconditionally energy-stable linear second-order finite difference scheme for Allen–Cahn equations

Cited by 19 publications

References 15 publications

Effective time step analysis for the Allen–Cahn equation with a high‐order polynomial free energy

Effective time step analysis for the Allen–Cahn equation with a high‐order polynomial free energy

Numerical analysis of a linear second-order finite difference scheme for space-fractional Allen–Cahn equations

Consistently and unconditionally energy-stable linear method for the diffuse-interface model of narrow volume reconstruction

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