# Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The Improved Moving Mesh PDEs

#### 2.1. Analysis of Huang’s MMPDEs

**Theorem**

**1.**

**Proof of Theorem**

**1.**

#### 2.2. The Moving Mesh Equation Based on the Characteristic Line

#### 2.3. The Proposed Moving Mesh Equations

**Remark**

**1.**

**Remark**

**2.**

## 3. The Numerical Algorithms

## 4. Experiments

#### 4.1. Advection-Diffusion Equation

#### 4.2. Burgers Equation

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Numerical solutions of Advection-diffusion equation applying Huang’s method (

**a**) and the proposed method (

**b**), N = 21.

**Figure 2.**The time step size for the Advection-diffusion equation with Huang’s method (

**a**) and the proposed method (

**b**), N = 21.

**Figure 3.**Numerical solutions and trajectories of the Burgers equation with $n=1$ applying Huang’s method (

**a**,

**b**) and the proposed method (

**c**,

**d**).

**Figure 4.**The time step sizes for Burgers equation with $n=1$ applying Huang’s method (

**a**) and the proposed method (

**b**).

**Figure 5.**Numerical solutions and trajectories of Burgers equation with $n=30$ applying Huang’s method (

**a**,

**b**) and proposed method (

**c**,

**d**).

**Figure 6.**The time step sizes for Burgers equation with $n=30$ applying Huang’s method (

**a**) and the proposed method (

**b**).

**Table 1.**The comparison of computational time and iterations of Huang’s method and the proposed method for different examples.

Huang’s method | The Proposed Method | |||
---|---|---|---|---|

Computational Time (s) | Iterations | Computational Time (s) | Iterations | |

Advection-diffusion equation | 2.47354 | 664 | 1.86607 | 279 |

Burgers equation with $n=1$ | 1.11280 | 258 | 0.89543 | 254 |

Burgers equation with $n=10$ | 4.6116 | 1236 | 2.43196 | 1154 |

Burgers equation with $n=20$ | 4.85784 | 2205 | 4.14323 | 2008 |

Burgers equation with $n=25$ | 8.48143 | 23,014 | 8.05290 | 8837 |

Burgers equation with $n=30$ | 70.785286 | 30,167 | 15.54634 | 13,053 |

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Gao, Q.; Zhang, S.
Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations. *Algorithms* **2016**, *9*, 86.
https://doi.org/10.3390/a9040086

**AMA Style**

Gao Q, Zhang S.
Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations. *Algorithms*. 2016; 9(4):86.
https://doi.org/10.3390/a9040086

**Chicago/Turabian Style**

Gao, Qinjiao, and Shenggang Zhang.
2016. "Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations" *Algorithms* 9, no. 4: 86.
https://doi.org/10.3390/a9040086