# An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation

^{*}

## Abstract

**:**

## 1. Introduction

**n**is the unit outer normal vector on the domain boundary $\partial \Omega $. The AC equation has been applied to phase transition, image processing, motion by mean curvature, multiphase flows, and dendritic growth (see e.g., [3,4,5,6,7,8] and the references therein). However, there are some mathematical problems that cannot be solved using the original form of the classical AC equation. For instance, the long range interactions in the interfacial dynamics is difficult to investigate using the original AC equation, therefore the AC equation with nonlocal diffusion (fractional) operator was analyzed in [9]. Meanwhile, the AC equation is not conservative, and Brassel and Bretin [10] proposed the nonlocal AC equation with the time-dependent Lagrange multiplier which preserves the shape of interface in local coordinates unlike the multiplier presented earlier in [11]. This equation has both nonlocal and local effects, even though the mass conservation property can be achieved in the local AC equation; hence we adopt the nonlocal equation in [10] and propose an explicit hybrid method in this paper for the following nonlocal AC equation with isotropically symmetric interfacial energy:

## 2. Numerical Algorithm

## 3. Numerical Results

#### 3.1. Evolution of Disks

#### 3.2. Comparison Test with a Conventional Method

#### 3.3. Cell Growth in a Complex Domain

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**(

**a**) Evolutions of the radii (${R}_{1}\left(t\right)$, ${R}_{2}\left(t\right)$, and ${R}_{3}\left(t\right)$) of three distinct disks at t = 0, 17,500$\Delta t$ and 22,150$\Delta t$. (

**b**) Comparison of the numerical solutions of three radii (star, circle, and diamond marks), the reference solution of three radii (solid line), and the reference solution of the average of ${R}_{1}\left(t\right)$ and ${R}_{3}\left(t\right)$ (dashed line).

**Figure 2.**(

**a**) Evolutions of the radii (${R}_{1}\left(t\right)$, ${R}_{2}\left(t\right)$, and ${R}_{3}\left(t\right)$) of three distinct disks at t = 0, 19,000$\Delta t$ and 22,500$\Delta t$. (

**b**) Comparison of the numerical solutions of three radii (star, circle, and diamond marks), the reference solution of three radii (solid line), and the reference solution of the average of ${R}_{1}\left(t\right)$ and ${R}_{3}\left(t\right)$ (dashed line).

**Figure 3.**(

**a**) Evolutions of the radii (${R}_{1}\left(t\right)$, ${R}_{2}\left(t\right)$, and ${R}_{3}\left(t\right)$) of three distinct disks with ${R}_{2}^{0}=0.18$ at t = 0, 13,000$\Delta t$, and 26,000$\Delta t$. (

**b**) Comparison of the numerical solutions of three radii (star, circle, and diamond marks), the reference solution of three radii (solid line), and the reference solution of the average of ${R}_{1}\left(t\right)$ and ${R}_{3}\left(t\right)$ (dashed line).

**Figure 4.**(

**a**) Evolutions of two disks with the explicit hybrid scheme at $t=0$ (dotted line), $2000\Delta t$ (dashed line), and $4000\Delta t$ (solid line). (

**b**) Evolution of two radii.

**Figure 6.**(

**a**–

**c**) Evolutions of cell growth in a star-shaped domain. (

**d**) Figure of the experiment in [28], which is reprinted with permission from Minc et al., Cell,

**144**, 414–426 (2011), ©2011, Elsevier.

**Figure 7.**(

**a**–

**c**) Evolutions of cell growth in a drop-shaped domain. (

**d**) Figure of the experiment in [28], which is reprinted with permission from Minc et al., Cell,

**144**, 414–426 (2011), ©2011, Elsevier.

**Table 1.**CPU times of the implicit hybrid method and ratios between the implicit hybrid scheme and the proposed scheme. CPU time when using the explicit hybrid method and $\Delta t=\Delta {t}_{\mathrm{ref}}$ is $43.946$ s.

$\Delta \mathit{t}$ | $\Delta {\mathit{t}}_{\mathbf{ref}}$ | $2\Delta {\mathit{t}}_{\mathbf{ref}}$ | $4\Delta {\mathit{t}}_{\mathbf{ref}}$ | $8\Delta {\mathit{t}}_{\mathbf{ref}}$ | $16\Delta {\mathit{t}}_{\mathbf{ref}}$ |
---|---|---|---|---|---|

CPU time (sec) | 198.452 | 94.984 | 48.499 | 29.000 | 17.124 |

Ratio | 4.40 | 2.16 | 1.10 | 0.66 | 0.39 |

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**MDPI and ACS Style**

Lee, C.; Yoon, S.; Park, J.; Kim, J.
An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation. *Symmetry* **2020**, *12*, 1218.
https://doi.org/10.3390/sym12081218

**AMA Style**

Lee C, Yoon S, Park J, Kim J.
An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation. *Symmetry*. 2020; 12(8):1218.
https://doi.org/10.3390/sym12081218

**Chicago/Turabian Style**

Lee, Chaeyoung, Sungha Yoon, Jintae Park, and Junseok Kim.
2020. "An Explicit Hybrid Method for the Nonlocal Allen–Cahn Equation" *Symmetry* 12, no. 8: 1218.
https://doi.org/10.3390/sym12081218