# Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method

## Abstract

**:**

## 1. Introduction

## 2. Swift–Hohenberg Type of Equation on a Narrow Band Domain

## 3. Numerical Method

## 4. Numerical Experiments

#### 4.1. Convergence Test

#### 4.2. Pattern Formation on a Sphere

#### 4.3. Pattern Formation on a Sphere Perturbed by a Spherical Harmonic

#### 4.4. Pattern Formation on a Spindle

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Evolution of $\varphi (x,y,z,t)$ with $g=0$. The yellow and blue regions indicate $\varphi =0.7540$ and $-0.7783$, respectively.

**Figure 2.**Evolution of $\varphi (x,y,z,t)$ with $g=1$. The yellow and blue regions indicate $\varphi =1.4320$ and $-0.7152$, respectively.

**Figure 3.**Evolution of $\mathcal{E}\left(\varphi \right)/\mathcal{E}\left({\varphi}^{0}\right)$ on the sphere with $g=0$ and 1.

**Figure 4.**Evolution of $\varphi (x,y,z,t)$ with $g=0$. The yellow and blue regions indicate $\varphi =0.8717$ and $-0.8372$, respectively.

**Figure 5.**Evolution of $\varphi (x,y,z,t)$ with $g=1$. The yellow and blue regions indicate $\varphi =1.4833$ and $-0.7135$, respectively.

**Figure 6.**Evolution of $\mathcal{E}\left(\varphi \right)/\mathcal{E}\left({\varphi}^{0}\right)$ on the perturbed sphere with $g=0$ and 1.

**Figure 7.**Evolution of $\varphi (x,y,z,t)$ with $g=0$. The yellow and blue regions indicate $\varphi =0.7059$ and $-0.7593$, respectively.

**Figure 8.**Evolution of $\varphi (x,y,z,t)$ with $g=1$. The yellow and blue regions indicate $\varphi =1.3842$ and $-0.6224$, respectively.

**Figure 9.**Evolution of $\mathcal{E}\left(\varphi \right)/\mathcal{E}\left({\varphi}^{0}\right)$ on the spindle with $g=0$ and 1.

$\mathsf{\Delta}\mathit{t}$ | $\mathit{T}/2$ | $\mathit{T}/{2}^{2}$ | $\mathit{T}/{2}^{3}$ | $\mathit{T}/{2}^{4}$ | |||
---|---|---|---|---|---|---|---|

${L}^{2}$-error | $5.445\times {10}^{-3}$ | $1.341\times {10}^{-3}$ | $2.862\times {10}^{-4}$ | $5.519\times {10}^{-5}$ | |||

Rate | 2.02 | 2.22 | 2.37 |

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Lee, H.G.
Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method. *Symmetry* **2019**, *11*, 1010.
https://doi.org/10.3390/sym11081010

**AMA Style**

Lee HG.
Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method. *Symmetry*. 2019; 11(8):1010.
https://doi.org/10.3390/sym11081010

**Chicago/Turabian Style**

Lee, Hyun Geun.
2019. "Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method" *Symmetry* 11, no. 8: 1010.
https://doi.org/10.3390/sym11081010