# Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation

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## Abstract

**:**

## 1. Introduction

## 2. Heat Equation

#### 2.1. Continuous Symmetry Groups

#### 2.2. Discrete Symmetry Group

## 3. Finite Difference Schemes for the Heat Equation

#### 3.1. Forward Difference Scheme

#### 3.2. Backward Difference Scheme

#### 3.3. Crank Nicolson Method

#### 3.4. Invariantization of the Crank Nicolson Method under the Discrete Symmetry Transformation

## 4. Discrete Symmetry Numerical Scheme For the Heat Equation

## 5. Solutions of the Heat Equation

#### 5.1. Analytic Solution

**Example**

**1.**

#### 5.2. Numerical Solutions Using CNM and the Proposed Method DCNM

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FTCS | Forward in Time and Centered in Space |

CNM | Crank Nicolson Method |

DCNM | Discritized Crank Nicolson Method |

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**Table 1.**Continuous symmetry transformations of (2).

Generators | Symmetry Transformations | |
---|---|---|

1 | $\frac{\partial}{\partial x}$ | Space translation: $(x,t,w)\mapsto (x+\u03f5,t,w)$ |

2 | $\frac{\partial}{\partial t}$ | Time translation: $(x,t,w)\mapsto (x,t+\u03f5,w)$ |

3 | $w\frac{\partial}{\partial w}$ | Scale Transformation: $(x,t,w)\mapsto (x,t,{e}^{\u03f5}w)$ |

4 | $x\frac{\partial}{\partial x}+2t\frac{\partial}{\partial t}$ | Scale Transformation: $(x,t,w)\mapsto ({e}^{\u03f5}x,{e}^{2\u03f5}t,w)$ |

5 | $2t\frac{\partial}{\partial x}-xw\frac{\partial}{\partial w}$ | Galilean boost: $(x,t,w)\mapsto (x+2\u03f5t,t,-xw\u03f5+w)$ |

6 | $4xt\frac{\partial}{\partial x}+4{t}^{2}\frac{\partial}{\partial t}-({x}^{2}+2t)w\frac{\partial}{\partial w}$ | Projection: $(x,t,w)\mapsto \phantom{\rule{4pt}{0ex}}\left(\frac{x}{1-4\u03f5t},\frac{t}{1-4\u03f5t},w\sqrt{1-4\u03f5t}exp\left(\frac{-\u03f5{x}^{2}}{1-4\u03f5t}\right)\right)$ |

7 | $q(x,t)\frac{\partial}{\partial w}$ |

${\mathit{x}}_{\mathit{i}}$ | FTCS | CNM | DCNM | Exact Solutions |
---|---|---|---|---|

0.0 | 0.000000 | 0.000000 | 0.0000000000 | 0.000000 |

0.2 | 0.000892 | 0.004517 | 0.0042956284 | 0.004227 |

0.4 | 0.001444 | 0.007309 | 0.00695047277 | 0.006840 |

0.6 | 0.001444 | 0.007309 | 0.00695047277 | 0.006840 |

0.8 | 0.000892 | 0.004517 | 0.0042956284 | 0.004227 |

1.0 | 0.000000 | 0.000000 | 0.0000000000 | 0.000000 |

${\mathit{x}}_{\mathit{i}}$ | FTCS | CNM | DCNM |
---|---|---|---|

0.0 | 0.000000 | 0.000000 | 0.0000000000 |

0.2 | 0.003335 | $2.9\times {10}^{-4}$ | $6.8345438\times {10}^{-5}$ |

0.4 | 0.005396 | $4.69\times {10}^{-4}$ | $1.10585242\times {10}^{-4}$ |

0.6 | 0.005396 | $4.69\times {10}^{-4}$ | $1.10585242\times {10}^{-4}$ |

0.8 | 0.003335 | $2.9\times {10}^{-4}$ | $6.8345438\times {10}^{-5}$ |

1.0 | 0.000000 | 0.000000 | 0.000000000000 |

${\mathit{x}}_{\mathit{i}}$ | $\mathit{k}=0.03$ | $\mathit{k}=0.02$ | $\mathit{k}=0.01$ |
---|---|---|---|

0.0 | 0.0000 | 0.0000 | 0.0000 |

0.2 | 0.0053 | 0.0045 | 0.0042 |

0.4 | 0.0085 | 0.0076 | 0.0069 |

0.6 | 0.0085 | 0.0076 | 0.0069 |

0.8 | 0.0053 | 0.0045 | 0.0042 |

1.0 | 0.0000 | 0.0000 | 0.0000 |

${\mathit{x}}_{\mathit{i}}$ | $\mathit{h}=0.3$ | $\mathit{h}=0.2$ | $\mathit{h}=0.1$ |
---|---|---|---|

0.0 | 0.0000 | 0.0000 | 0.0000 |

0.2 | 0.0050 | 0.0047 | 0.0045 |

0.4 | 0.0081 | 0.0078 | 0.0076 |

0.6 | 0.0081 | 0.0078 | 0.0076 |

0.8 | 0.0050 | 0.0047 | 0.0045 |

1.0 | 0.0000 | 0.0000 | 0.0000 |

t | ${\mathit{L}}_{2}$ DCNM | ${\mathit{L}}_{\mathit{\infty}}$ DCNM | ${\mathit{L}}_{2}$ CNM | ${\mathit{L}}_{\mathit{\infty}}$ CNM |
---|---|---|---|---|

0.1 | $9.6\times {10}^{-3}$ | $4.3\times {10}^{-3}$ | $4.86\times {10}^{-3}$ | $6.79\times {10}^{-3}$ |

0.3 | $4.0\times {10}^{-4}$ | $1.8\times {10}^{-4}$ | $8.87\times {10}^{-5}$ | $3.76\times {10}^{-4}$ |

0.5 | $9.0830\times {10}^{-4}$ | $4.0812\times {10}^{-4}$ | $1.73\times {10}^{-3}$ | $2.44\times {10}^{-4}$ |

0.7 | $1.8024\times {10}^{-4}$ | $1.0097\times {10}^{-4}$ | $2.04\times {10}^{-4}$ | $3.17\times {10}^{-4}$ |

0.9 | $1.0224\times {10}^{-4}$ | $6.1223\times {10}^{-5}$ | $2.14\times {10}^{-3}$ | $3.14\times {10}^{-3}$ |

1 | $1.1556\times {10}^{-4}$ | $5.0808\times {10}^{-5}$ | $2.15\times {10}^{-3}$ | $3.32\times {10}^{-3}$ |

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**MDPI and ACS Style**

Bibi, K.; Feroze, T.
Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation. *Symmetry* **2020**, *12*, 359.
https://doi.org/10.3390/sym12030359

**AMA Style**

Bibi K, Feroze T.
Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation. *Symmetry*. 2020; 12(3):359.
https://doi.org/10.3390/sym12030359

**Chicago/Turabian Style**

Bibi, Khudija, and Tooba Feroze.
2020. "Discrete Symmetry Group Approach for Numerical Solution of the Heat Equation" *Symmetry* 12, no. 3: 359.
https://doi.org/10.3390/sym12030359