#
A Linear, Second-Order, and Unconditionally Energy-Stable Method for the L^{2}-Gradient Flow-Based Phase-Field Crystal Equation

## Abstract

**:**

## 1. Introduction

## 2. Linear, Second-Order, and Unconditionally Energy-Stable Method

**Lemma**

**1.**

**Proof.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Numerical Experiments

#### 3.1. Accuracy Test

#### 3.2. Energy Stability Test

#### 3.3. Pattern Formation

#### 3.4. Crystal Growth

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Evolution of $\mathcal{F}\left(t\right)$ for the reference solution with $\u03f5=0.025$, $\Delta x=\Delta y=\frac{1}{3}$, and $\Delta t={2}^{-10}$. (

**b**) Relative ${l}_{2}$-errors of $\varphi (x,y,1)$ for $\Delta t={2}^{-8},{2}^{-7},\dots ,1$. (

**c**) Evolution of ${\int}_{\Omega}(\varphi (x,y,t)-\varphi (x,y,0))\phantom{\rule{0.222222em}{0ex}}dxdy$ for various time steps.

**Figure 2.**(

**a**) Evolution of $\mathcal{F}\left(t\right)$ with several time steps. (

**b**) Evolution of $\varphi (x,y,t)$ with $\u03f5=0.25$, $\Delta x=\Delta y=\frac{1}{3}$, and $\Delta t={2}^{-4}$. The yellow, green, and blue regions show $\varphi =0.6288$, $0.0688$, and $-0.4913$, respectively.

**Figure 3.**Evolution of $\varphi (x,y,t)$ using the operator splitting method in [17] with (

**a**) $\overline{\varphi}=0.02$ and (

**b**) $0.2$. Here, $\u03f5=0.2$, $\Delta x=\Delta y=\frac{1}{3}$, and $\Delta t=2$ are used. The yellow, green, and blue regions show $\varphi =0.5$, 0, and $-0.5$, respectively.

**Figure 4.**Evolution of $\varphi (x,y,t)$ using the proposed method with (

**a**) $\overline{\varphi}=0.02$ and (

**b**) $0.2$. Here, $\u03f5=0.2$, $\Delta x=\Delta y=\frac{1}{3}$, and $\Delta t=2$ are used. The yellow, green, and blue regions show $\varphi =0.5$, 0, and $-0.5$, respectively.

**Figure 5.**Evolution of $\mathcal{F}\left(t\right)$ and $\varphi (x,y,t)$ with $\u03f5=0.25$, $\Delta x=\Delta y=1$, and $\Delta t=1$. The red, green, and blue regions show $\varphi =0.6475$, $0.0741$, and $-0.4993$, respectively.

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**MDPI and ACS Style**

Lee, H.G.
A Linear, Second-Order, and Unconditionally Energy-Stable Method for the *L*^{2}-Gradient Flow-Based Phase-Field Crystal Equation. *Mathematics* **2022**, *10*, 548.
https://doi.org/10.3390/math10040548

**AMA Style**

Lee HG.
A Linear, Second-Order, and Unconditionally Energy-Stable Method for the *L*^{2}-Gradient Flow-Based Phase-Field Crystal Equation. *Mathematics*. 2022; 10(4):548.
https://doi.org/10.3390/math10040548

**Chicago/Turabian Style**

Lee, Hyun Geun.
2022. "A Linear, Second-Order, and Unconditionally Energy-Stable Method for the *L*^{2}-Gradient Flow-Based Phase-Field Crystal Equation" *Mathematics* 10, no. 4: 548.
https://doi.org/10.3390/math10040548