A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes

Hwang, Yeong‐Maw; Ham, Seokjun; Lee, Chaeyoung; Lee, Gyeonggyu; Kang, Shinjae; Kim, Junseok

doi:10.3934/era.2023233

era

2023

DOI: 10.3934/era.2023233

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A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes

Yeong‐Maw Hwang¹,

Seokjun Ham²,

Chaeyoung Lee³

et al.

Abstract: <abstract><p>In this paper, we propose a novel, simple, efficient, and explicit numerical method for the Allen–Cahn (AC) equation on effective symmetric triangular meshes. First, we compute the net vector of all vectors starting from each node point to its one-ring neighbor vertices and virtually adjust the neighbor vertices so that the net vector is zero. Then, we define the values at the virtually adjusted nodes using linear and quadratic interpolations. Finally, we define a discrete Laplace oper… Show more

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2024

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Cited by 3 publications

References 27 publications

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A modified Allen–Cahn equation with a mesh size-dependent interfacial parameter on a triangular mesh

Yang,

Wang,

Kwak

et al. 2024

Computer Physics Communications

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A modified Allen–Cahn equation with a mesh size-dependent interfacial parameter on a triangular mesh

Yang,

Wang,

Kwak

et al. 2024

Computer Physics Communications

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Modified Wave-Front Propagation and Dynamics Coming from Higher-Order Double-Well Potentials in the Allen–Cahn Equations

Kim

2024

Mathematics

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In this paper, we conduct a numerical investigation into the influence of polynomial order on wave-front propagation in the Allen–Cahn (AC) equations with high-order polynomial potentials. The conventional double-well potential in these equations is typically a fourth-order polynomial. However, higher-order double-well potentials, such as sixth, eighth, or any even order greater than four, can model more complex dynamics in phase transition problems. Our study aims to explore how the order of these polynomial potentials affects the speed and behavior of front propagation in the AC framework. By systematically varying the polynomial order, we observe significant changes in front dynamics. Higher-order polynomials tend to influence the sharpness and speed of moving fronts, leading to modifications in the overall pattern formation process. These results have implications for understanding the role of polynomial potentials in phase transition phenomena and offer insights into the broader application of AC equations for modeling complex systems. This work demonstrates the importance of considering higher-order polynomial potentials when analyzing front propagation and phase transitions, as the choice of polynomial order can dramatically alter system behavior.

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Numerical investigation of the dynamics for a normalized time-fractional diffusion equation

Lee,

Nam,

Bang

et al. 2024

MATH

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<p>In this study, we proposed a normalized time-fractional diffusion equation and conducted a numerical investigation of the dynamics of the proposed equation. We discretized the governing equation by using a finite difference method. The proposed normalized time-fractional diffusion equation features a different time scale compared to the conventional time-fractional diffusion equation. This distinct time scale provides an intuitive understanding of the fractional time derivative, which represents a weighted average of the temporal history of the time derivative. Furthermore, the sum of the weight function is one for all values of the fractional parameter and time. The primary advantage of the proposed model over conventional time-fractional equations is the unity property of the sum of the weight function, which allows us to investigate the effects of the fractional order on the evolutionary dynamics of time-fractional equations. To highlight the differences in performance between the conventional and normalized time-fractional diffusion equations, we have conducted several numerical experiments.</p>

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A simple and efficient numerical method for the Allen–Cahn equation on effective symmetric triangular meshes

Cited by 3 publications

References 27 publications

A modified Allen–Cahn equation with a mesh size-dependent interfacial parameter on a triangular mesh

A modified Allen–Cahn equation with a mesh size-dependent interfacial parameter on a triangular mesh

Modified Wave-Front Propagation and Dynamics Coming from Higher-Order Double-Well Potentials in the Allen–Cahn Equations

Numerical investigation of the dynamics for a normalized time-fractional diffusion equation

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