# A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Construction of PASE-I Parallel Difference Scheme

## 3. Theoretical Analysis of PASE-I Difference Scheme

#### 3.1. The Existence and Uniqueness of PASE-I Scheme’s Solution

**Lemma**

**1.**

**Proof.**

**Theorem**

**1.**

#### 3.2. Stability of PASE-I Scheme

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

#### 3.3. Convergence of PASE-I Scheme

**Theorem**

**3.**

## 4. PASI-E Parallel Difference Scheme

**Theorem**

**4.**

## 5. Numerical Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Distribution of the difference total energy in spatial grid points (M takes $50,100,200,400$).

**Table 1.**${E}_{\infty}$ and $Orde{r}_{x}$ of PASE-I and PASI-E schemes $({\tau}^{\alpha}\approx {h}^{2})$.

${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | M | N | PASE-I Scheme | PASI-E Scheme | ||
---|---|---|---|---|---|---|---|

${\mathit{E}}_{\infty}$ | ${\mathit{Order}}_{\mathit{x}}$ | ${\mathit{E}}_{\infty}$ | ${\mathit{Order}}_{\mathit{x}}$ | ||||

0.4 | 0.4 | 50 | 132 | 1.960988$\times {10}^{-3}$ | —— | 1.933065$\times {10}^{-3}$ | —— |

100 | 316 | 4.750145$\times {10}^{-4}$ | 2.045537 | 4.715343$\times {10}^{-4}$ | 2.035455 | ||

200 | 752 | 1.175639$\times {10}^{-4}$ | 2.014526 | 1.171268$\times {10}^{-4}$ | 2.009290 | ||

400 | 1788 | 2.927383$\times {10}^{-5}$ | 2.005761 | 2.921901$\times {10}^{-5}$ | 2.003092 | ||

800 | 4254 | 7.293185$\times {10}^{-6}$ | 2.004990 | 7.286329$\times {10}^{-6}$ | 2.003643 | ||

0.5 | 0.5 | 50 | 184 | 1.808959$\times {10}^{-3}$ | —— | 1.783177$\times {10}^{-3}$ | —— |

100 | 464 | 4.441176$\times {10}^{-4}$ | 2.026146 | 4.408679$\times {10}^{-4}$ | 2.016032 | ||

200 | 1169 | 1.093836$\times {10}^{-4}$ | 2.021544 | 1.097908$\times {10}^{-4}$ | 2.005589 | ||

400 | 2947 | 2.716023$\times {10}^{-5}$ | 2.009829 | 2.721104$\times {10}^{-5}$ | 2.012493 | ||

800 | 7426 | 6.763548$\times {10}^{-6}$ | 2.005643 | 6.757212$\times {10}^{-6}$ | 2.009691 | ||

0.6 | 0.6 | 50 | 267 | 1.344353$\times {10}^{-4}$ | —— | 1.345261$\times {10}^{-4}$ | —— |

100 | 719 | 3.270205$\times {10}^{-5}$ | 2.039459 | 3.271047$\times {10}^{-5}$ | 2.040061 | ||

200 | 1937 | 8.019243$\times {10}^{-6}$ | 2.027843 | 8.020099$\times {10}^{-6}$ | 2.028060 | ||

400 | 5214 | 1.979933$\times {10}^{-6}$ | 2.018014 | 1.979839$\times {10}^{-6}$ | 2.018236 | ||

800 | 14036 | 4.901814$\times {10}^{-7}$ | 2.014064 | 4.901704$\times {10}^{-7}$ | 2.014027 |

${\mathit{\alpha}}_{1}$ | ${\mathit{\alpha}}_{2}$ | N | PASE-I Scheme | PASI-E Scheme | ||
---|---|---|---|---|---|---|

${\mathit{E}}_{\infty}$ | ${\mathit{Order}}_{\mathit{t}}$ | ${\mathit{E}}_{\infty}$ | ${\mathit{Order}}_{\mathit{t}}$ | |||

0.35 | 0.35 | 200 | 1.540090$\times {10}^{-4}$ | —— | 1.541745$\times {10}^{-4}$ | —— |

400 | 4.981818$\times {10}^{-5}$ | 1.628270 | 4.986837$\times {10}^{-5}$ | 1.628367 | ||

800 | 1.579056$\times {10}^{-5}$ | 1.657609 | 1.580609$\times {10}^{-5}$ | 1.657643 | ||

1600 | 4.971665$\times {10}^{-6}$ | 1.667262 | 4.976502$\times {10}^{-6}$ | 1.667277 | ||

3200 | 1.561734$\times {10}^{-6}$ | 1.670579 | 1.563244$\times {10}^{-6}$ | 1.670588 | ||

0.4 | 0.2 | 200 | 1.641599$\times {10}^{-4}$ | —— | 1.643375$\times {10}^{-4}$ | —— |

400 | 5.320684$\times {10}^{-5}$ | 1.625418 | 5.326029$\times {10}^{-5}$ | 1.625530 | ||

800 | 1.681644$\times {10}^{-5}$ | 1.661739 | 1.683288$\times {10}^{-5}$ | 1.661777 | ||

1600 | 5.266880$\times {10}^{-6}$ | 1.674851 | 5.271969$\times {10}^{-6}$ | 1.674868 | ||

3200 | 1.643123$\times {10}^{-6}$ | 1.680508 | 1.644700$\times {10}^{-6}$ | 1.680517 | ||

0.8 | 0.2 | 200 | 6.070178$\times {10}^{-4}$ | —— | 6.075953$\times {10}^{-4}$ | —— |

400 | 2.349702$\times {10}^{-4}$ | 1.369260 | 2.351861$\times {10}^{-4}$ | 1.369307 | ||

800 | 9.139187$\times {10}^{-5}$ | 1.362340 | 9.147204$\times {10}^{-5}$ | 1.362400 | ||

1600 | 3.598744$\times {10}^{-5}$ | 1.344572 | 3.601678$\times {10}^{-5}$ | 1.344661 | ||

3200 | 1.445759$\times {10}^{-5}$ | 1.315666 | 1.446810$\times {10}^{-5}$ | 1.315793 |

100 | 500 | 1000 | 2000 | 3000 | 4000 | 5000 | |
---|---|---|---|---|---|---|---|

Implicit | 9.90432 | 53.6988 | 139.582 | 359.223 | 499.958 | 900.628 | 1462.45 |

PASE-I | 9.14516 | 44.5909 | 91.2816 | 171.389 | 272.927 | 368.622 | 493.003 |

PASI-E | 10.5672 | 44.7052 | 91.4469 | 171.517 | 262.139 | 368.364 | 491.059 |

${S}_{P}$ of PASE-I | 1.08301 | 1.20425 | 1.52914 | 2.09594 | 1.83183 | 2.44322 | 2.96641 |

${S}_{P}$ of PASI-E | 0.93726 | 1.20117 | 1.52637 | 2.09437 | 1.90722 | 2.44494 | 2.97815 |

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Yang, X.; Wu, L.
A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model. *Mathematics* **2020**, *8*, 596.
https://doi.org/10.3390/math8040596

**AMA Style**

Yang X, Wu L.
A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model. *Mathematics*. 2020; 8(4):596.
https://doi.org/10.3390/math8040596

**Chicago/Turabian Style**

Yang, Xiaozhong, and Lifei Wu.
2020. "A New Kind of Parallel Natural Difference Method for Multi-Term Time Fractional Diffusion Model" *Mathematics* 8, no. 4: 596.
https://doi.org/10.3390/math8040596