# Numerical Simulation of Partial Differential Equations via Local Meshless Method

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Implementation of Numerical Method

#### Local Meshless Method for KdVB Equation

## 3. Time Discretization

#### A $\theta $-Weighted Technique for 2D Diffusion Equation

## 4. Stability Analysis

## 5. Numerical Analysis

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

**Problem**

**4.**

**Problem**

**5.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**$\rho (W)$ versus $dt$ for the explicit scheme (

**left**) and Implicit and Crank–Nicolson scheme (

**right**) with $N=21$ for Problem 3.

**Figure 2.**$Re(\lambda )$ versus c for the global meshless method (GMM) (

**left**) and the local meshless method (LMM) (

**right**) for Problem 3.

**Figure 3.**Matrix structure of the LMM of local sub-domain three in the 1D case (

**left**), local sub-domain five in the 2D case (

**right**).

**Figure 4.**Numerical solution of the Kortewege–de Vries–Burgers’ (KdVB) equation for $\beta =0.001$ (

**left**) $\beta =0.1$ (

**right**), for Problem 1.

**Figure 5.**Numerical solution of the KdVB equation for $\beta =0.5$ (

**left**) $\beta =1$ (

**right**), for Problem 1.

**Table 1.**Comparison of the numerical solution of the Kortewege–de Vries–Burgers’ (KdVB) equation for Problem 1.

t | ${\mathit{L}}_{\mathbf{\infty}}$ LMM | ${\mathit{L}}_{2}$ LMM | ${\mathit{L}}_{\mathbf{\infty}}$ [5] | ${\mathit{L}}_{2}$ [5] |
---|---|---|---|---|

$\beta =0.004$ | ||||

1 | 5.5555 × 10${}^{-12}$ | 2.8402 × 10${}^{-10}$ | 6.822 × 10${}^{-0}$ | 8.845 × 10${}^{-9}$ |

2 | 1.1110 × 10${}^{-11}$ | 5.6792 × 10${}^{-10}$ | 1.150 × 10${}^{-8}$ | 1.652 × 10${}^{-8}$ |

3 | 1.6661 × 10${}^{-11}$ | 8.5171 × 10${}^{-10}$ | 1.485 × 10${}^{-8}$ | 2.338 × 10${}^{-8}$ |

10 | 5.5441 × 10${}^{-11}$ | 2.8353 × 10${}^{-9}$ | 2.479 × 10${}^{-8}$ | 6.046 × 10${}^{-8}$ |

$\beta =0.04$ | ||||

1 | 8.4705 × 10${}^{-8}$ | 1.5103 × 10${}^{-6}$ | 2.936 × 10${}^{-6}$ | 3.727 × 10${}^{-7}$ |

2 | 1.6937 × 10${}^{-7}$ | 3.0211 × 10${}^{-6}$ | 4.204 × 10${}^{-6}$ | 2.207 × 10${}^{-8}$ |

3 | 2.5400 × 10${}^{-7}$ | 4.5322 × 10${}^{-6}$ | 4.126 × 10${}^{-6}$ | 1.928 × 10${}^{-6}$ |

10 | 8.4535 × 10${}^{-7}$ | 1.5111 × 10${}^{-5}$ | 5.800 × 10${}^{-6}$ | 1.297 × 10${}^{-5}$ |

$\beta =0.1$ | ||||

1 | 7.7105 × 10${}^{-6}$ | 1.1800 × 10${}^{-5}$ | 1.540 × 10${}^{-5}$ | 1.004 × 10${}^{-5}$ |

2 | 1.5374 × 10${}^{-5}$ | 2.3912 × 10${}^{-5}$ | 3.076 × 10${}^{-5}$ | 1.732 × 10${}^{-5}$ |

3 | 2.2987 × 10${}^{-5}$ | 3.6284 × 10${}^{-5}$ | 4.604 × 10${}^{-5}$ | 2.874 × 10${}^{-5}$ |

10 | 7.4772 × 10${}^{-5}$ | 1.2861 × 10${}^{-4}$ | 1.498 × 10${}^{-4}$ | 1.342 × 10${}^{-4}$ |

t | ${\mathit{L}}_{\mathbf{\infty}}$ | ${\mathit{L}}_{2}$ | CPU Time | ${\mathit{L}}_{\mathbf{\infty}}$ | ${\mathit{L}}_{2}$ |
---|---|---|---|---|---|

LMM | LMM | LMM | [8] | [8] | |

1 | 2.8316 × 10${}^{-10}$ | 3.1032 × 10${}^{-8}$ | 0.26 | 2.06127 × 10${}^{-6}$ | 4.62479 × 10${}^{-6}$ |

3 | 8.4947 × 10${}^{-10}$ | 9.3095 × 10${}^{-8}$ | 0.72 | 1.14705 × 10${}^{-6}$ | 9.14997 × 10${}^{-7}$ |

5 | 1.4158 × 10${}^{-9}$ | 1.5515 × 10${}^{-7}$ | 1.15 | 1.28565 × 10${}^{-6}$ | 1.21327 × 10${}^{-6}$ |

7 | 1.9821 × 10${}^{-9}$ | 2.1721 × 10${}^{-7}$ | 1.63 | 1.51656 × 10${}^{-6}$ | 1.45490 × 10${}^{-6}$ |

9 | 2.5484 × 10${}^{-9}$ | 2.7927 × 10${}^{-7}$ | 2.18 | 1.67755 × 10${}^{-6}$ | 1.66392 × 10${}^{-6}$ |

${\mathit{L}}_{\mathbf{\infty}}$ | ||||||
---|---|---|---|---|---|---|

x = 0.1 | x = 0.3 | x = 0.5 | ||||

t | LMM | [11] | LMM | [11] | LMM | [11] |

0.1 | 4.44 × 10${}^{-16}$ | 4.5230 × 10${}^{-13}$ | 5.55 × 10${}^{-17}$ | 1.3291 × 10${}^{-12}$ | 6.10 × 10${}^{-16}$ | 6.6992 × 10${}^{-12}$ |

0.2 | 2.16 × 10${}^{-15}$ | 1.3524 × 10${}^{-12}$ | 1.22 × 10${}^{-15}$ | 1.3696 × 10${}^{-12}$ | 1.11 × 10${}^{-16}$ | 2.1989 × 10${}^{-12}$ |

0.3 | 5.21 × 10${}^{-15}$ | 2.2525 × 10${}^{-12}$ | 3.83 × 10${}^{-15}$ | 4.0690 × 10${}^{-12}$ | 2.16 × 10${}^{-15}$ | 2.3009 × 10${}^{-12}$ |

0.4 | 9.60 × 10${}^{-15}$ | 3.1523 × 10${}^{-12}$ | 7.71 × 10${}^{-15}$ | 6.7682 × 10${}^{-12}$ | 5.55 × 10${}^{-15}$ | 6.8008 × 10${}^{-12}$ |

0.5 | 1.53 × 10${}^{-14}$ | 4.0521 × 10${}^{-12}$ | 1.29 × 10${}^{-14}$ | 9.4666 × 10${}^{-12}$ | 1.03 × 10${}^{-14}$ | 1.1300 × 10${}^{-11}$ |

${\mathit{L}}_{\mathbf{\infty}}$ | ||||||
---|---|---|---|---|---|---|

x = 0.1 | x = 0.3 | x = 0.5 | ||||

t | LMM | [11] | LMM | [11] | LMM | [11] |

0.1 | 7.81 × 10${}^{-12}$ | 2.2379 × 10${}^{-11}$ | 2.30 × 10${}^{-11}$ | 6.9696 × 10${}^{-11}$ | 3.82 × 10${}^{-11}$ | 1.2050 × 10${}^{-10}$ |

0.2 | 1.60 × 10${}^{-11}$ | 4.4328 × 10${}^{-11}$ | 4.64 × 10${}^{-11}$ | 1.3552 × 10${}^{-10}$ | 7.69 × 10${}^{-11}$ | 2.3025 × 10${}^{-10}$ |

0.3 | 2.46 × 10${}^{-11}$ | 6.6276 × 10${}^{-11}$ | 7.03 × 10${}^{-11}$ | 2.0134 × 10${}^{-10}$ | 1.15 × 10${}^{-10}$ | 3.3999 × 10${}^{-10}$ |

0.4 | 3.37 × 10${}^{-11}$ | 8.8225 × 10${}^{-11}$ | 9.46 × 10${}^{-11}$ | 2.6716 × 10${}^{-10}$ | 1.55 × 10${}^{-10}$ | 4.4973 × 10${}^{-10}$ |

0.5 | 4.31 × 10${}^{-11}$ | 1.1017 × 10${}^{-10}$ | 1.19 × 10${}^{-10}$ | 3.3298 × 10${}^{-10}$ | 1.95 × 10${}^{-10}$ | 5.5947 × 10${}^{-10}$ |

**Table 5.**Numerical solution for a wide range of the shape parameter value c for $\alpha =0.0001$ and $k=0.01$ for Problem 2.

c | U | V | c | U | V |
---|---|---|---|---|---|

0.001 | 1.4629 × 10${}^{-12}$ | 3.0635 × 10${}^{-9}$ | 40 | 6.0559 × 10${}^{-13}$ | 1.2333 × 10${}^{-9}$ |

0.01 | 1.1800 × 10${}^{-11}$ | 2.0888 × 10${}^{-7}$ | 80 | 5.6560 × 10${}^{-13}$ | 1.1480 × 10${}^{-9}$ |

0.1 | 1.1655 × 10${}^{-11}$ | 2.2179 × 10${}^{-7}$ | 160 | 5.5559 × 10${}^{-13}$ | 1.1268 × 10${}^{-9}$ |

1 | 1.1747 × 10${}^{-11}$ | 2.0555 × 10${}^{-7}$ | 320 | 5.5265 × 10${}^{-13}$ | 1.1216 × 10${}^{-9}$ |

5 | 3.8443e × 10${}^{-12}$ | 1.0146 × 10${}^{-8}$ | 640 | 5.9097 × 10${}^{-13}$ | 1.1239 × 10${}^{-9}$ |

10 | 1.4029 × 10${}^{-12}$ | 3.0469 × 10${}^{-9}$ | 1000 | 9.0130 × 10${}^{-13}$ | 1.1524 × 10${}^{-9}$ |

20 | 7.6552 × 10${}^{-13}$ | 1.5794 × 10${}^{-9}$ | 1200 | 6.1639 × 10${}^{-13}$ | 1.1081 × 10${}^{-9}$ |

**Table 6.**Numerical solution for the range of the shape parameter value c for $\alpha =0.0001$ and $k=0.01$ for Problem 2.

$\mathit{c}$ | 0.001 | 0.01 | 0.1 | 0.2 | 0.3 |
---|---|---|---|---|---|

U | 2.8405 × 10${}^{-10}$ | 3.4306 × 10${}^{-9}$ | 3.5230 × 10${}^{-10}$ | 1.4130 × 10${}^{-10}$ | ⋯ |

V | 5.2746 × 10${}^{-7}$ | 6.5458 × 10${}^{-6}$ | 9.1756 × 10${}^{-7}$ | 3.5451 × 10${}^{-}7$ | ⋯ |

t | 1 | 5 | 10 | 15 | 20 |
---|---|---|---|---|---|

${L}_{\infty}$ | 6.8814 × 10${}^{-4}$ | 3.4458 × 10${}^{-3}$ | 7.0255 × 10${}^{-3}$ | 1.2374 × 10${}^{-2}$ | 2.0904 × 10${}^{-2}$ |

CPU time (in s) | 2.54 | 2.59 | 2.71 | 2.85 | 2.88 |

N | P(1,1,1) | P(1.1,1.1,1.1) | CPU Time |
---|---|---|---|

$8\times 8\times 8$ | 0.051746 | 0.0278333 | 7.6 |

$12\times 12\times 12$ | 0.055708 | 0.021232 | 32.7 |

$16\times 16\times 16$ | 0.058240 | 0.020440 | 128.9 |

$20\times 20\times 20$ | 0.059639 | 0.020797 | 554.9 |

N | ${\mathit{L}}_{\mathit{abs}}$ at P(1,1,1) | ${\mathit{L}}_{\mathit{abs}}$ at P(1.1,1.1,1.1) |
---|---|---|

$4\times 4\times 4$ | 9.7425 × 10${}^{-3}$ | 2.0440 × 10${}^{-2}$ |

$8\times 8\times 8$ | 6.4939 × 10${}^{-3}$ | 7.3929 × 10${}^{-3}$ |

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**MDPI and ACS Style**

Ahmad, I.; Riaz, M.; Ayaz, M.; Arif, M.; Islam, S.; Kumam, P.
Numerical Simulation of Partial Differential Equations via Local Meshless Method. *Symmetry* **2019**, *11*, 257.
https://doi.org/10.3390/sym11020257

**AMA Style**

Ahmad I, Riaz M, Ayaz M, Arif M, Islam S, Kumam P.
Numerical Simulation of Partial Differential Equations via Local Meshless Method. *Symmetry*. 2019; 11(2):257.
https://doi.org/10.3390/sym11020257

**Chicago/Turabian Style**

Ahmad, Imtiaz, Muhammad Riaz, Muhammad Ayaz, Muhammad Arif, Saeed Islam, and Poom Kumam.
2019. "Numerical Simulation of Partial Differential Equations via Local Meshless Method" *Symmetry* 11, no. 2: 257.
https://doi.org/10.3390/sym11020257