# Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

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## Abstract

**:**

## 1. Introduction

## 2. Implementation of the Numerical Method

#### 2.1. Implementation of LMM for the KG Equation

#### 2.2. Implementation of LMM for the 2D Model Equations

**f**is a vector of the corresponding initial condition of the problem. Orders of the vectors $\mathbf{h}$ and

**f**are ${N}^{n}\times 1$, where $n=1;2$ for one- and two-dimensional PDEs, respectively.

## 3. Numerical Analysis

**Test**

**Problem 1.**

**Test**

**Problem 2.**

**Test**

**Problem 3.**

**Test**

**Problem 4.**

**Test**

**Problem 5.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wazwaz, M.A. New travelling wave solutions to the Boussinesq and the Klein–Gordon equations. Commun. Nonlinear Sci. Numer. Simul.
**2008**, 13, 889–901. [Google Scholar] [CrossRef] - El-Sayed, S.M. The decomposition method for studying the Klein–Gordon equation. Chaos Soliton Fract.
**2003**, 18, 1025–1030. [Google Scholar] [CrossRef] - Duncan, D.B. Symplectic finite difference approximations of the nonlinear Klein–Gordon equation. SIAM J. Numer. Anal.
**1997**, 34, 1742–1760. [Google Scholar] [CrossRef] - Lee, I.J. Numerical solution for nonlinear Klein–Gordon equation by collocation method with respect to spectral method. J. Korean Math. Soc.
**1995**, 32, 541–551. [Google Scholar] - Dehghan, M.; Shokri, A. Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions. J. Comput. Appl. Math.
**2009**, 230, 400–410. [Google Scholar] [CrossRef] [Green Version] - Lakestani, M.; Dehghan, M. Collocation and finite difference-collocation methods for the solution of nonlinear Klein–Gordon equation. Comput. Phys. Commun.
**2010**, 181, 1392–1401. [Google Scholar] [CrossRef] - Pekmen, B.; Tezer-Sezgin, M. Differential quadrature solution of nonlinear Klein–Gordon and Sine-Gordon equations. Comput. Phys. Commun.
**2012**, 183, 1702–1713. [Google Scholar] [CrossRef] - Hariharan, G. Haar wavelet method for solving the Klein–Gordon and the Sine-Gordon equations. Int. J. Nonlinear Sci.
**2011**, 11, 180–189. [Google Scholar] - Yin, F.; Song, J.; Lu, F. A coupled method of Laplace transform and Legendre wavelets for nonlinear Klein–Gordon equations. Math. Methods Appl. Sci.
**2014**, 37, 781–791. [Google Scholar] [CrossRef] - Li, Q.; Ji, Z.; Zheng, Z.; Liu, H. Numerical solution of nonlinear Klein–Gordon equation using Lattice Boltzmann method. Appl. Math.
**2011**, 2, 1479–1485. [Google Scholar] [CrossRef] - Sarboland, M.; Aminataei, A. Numerical solution of the nonlinear Klein–Gordon equation using multiquadric quasi-interpolation scheme. Univ. J. Math. Appl.
**2015**, 3, 40–49. [Google Scholar] - Bahadir, A.R. A fully implicit finite difference scheme for two-dimensional Burgers’ equations. Appl. Math. Comput.
**2003**, 137, 131–137. [Google Scholar] [CrossRef] - Younga, D.; Fana, C.; Hua, S.; Atluri, S. The Eulerian-Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations. Eng. Anal. Bound. Elem.
**2009**, 32, 395–412. [Google Scholar] [CrossRef] - Ali, A.; Siraj-ul-Islam; Haq, S. A computational meshfree technique for the numerical solution of the two-dimensional coupled Burgers’ equations. Int. J. Comput. Meth. Eng. Sci. Mech.
**2009**, 10, 406–412. [Google Scholar] [CrossRef] - Zhang, X.; Zhu, H.; Kuo, L. A comparison study of the LMAPS method and the LDQ method for time-dependent problems. Eng. Anal. Bound. Elem.
**2013**, 37, 1408–1415. [Google Scholar] [CrossRef] - Dehghan, M.; Salehi, R. The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas. Comput. Phys. Commun.
**2011**, 182, 2540–2549. [Google Scholar] [CrossRef] - Dehghana, M.; Abbaszadeha, M.; Mohebbib, A. The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin-Bona-Mahony-Burgers and regularized long-wave equations on non-rectangular domains with error estimate. J. Comput. Appl. Math.
**2015**, 286, 211–231. [Google Scholar] [CrossRef] - Abdulloev, K.O.; Bogolubsky, I.L.; Makhankov, V.G. One more example of inelastic soliton interaction. Phys. Lett.
**1976**, 56A, 427–428. [Google Scholar] [CrossRef] - Bona, J.L.; Bryant, P.J. A mathematical model for long waves generated by wave makers in nonlinear dispersive systems. Proc. Camb. Philos. Soc.
**1973**, 73, 391–405. [Google Scholar] [CrossRef] - Bhardwaj, D.; Shankar, R. A computational method for regularized long wave equation. Comput. Math. Appl.
**2000**, 40, 1397–1404. [Google Scholar] [CrossRef] [Green Version] - Esen, A.; Kutluay, S. Application of a lumped Galerkin method to the regularized long wave equation. Appl. Math. Comput.
**2006**, 174, 833–845. [Google Scholar] [CrossRef] - Gou, B.Y.; Cao, W.M. The Fourier pseudospectral method with a restrain operator for the RLW equation. Comput. Math. Appl.
**1988**, 74, 110–126. [Google Scholar] - Raslan, K.R. A computational method for the regularized long wave equation. Appl. Math. Comput.
**2005**, 167, 1101–1118. [Google Scholar] [CrossRef] - Roshan, T. A Petrov-Galerkin method for solving the generalized regularized long wave (GRLW) equation. Comput. Math. Appl.
**2012**, 63, 943–956. [Google Scholar] [CrossRef] - Guo, P.F.; Zhang, L.W.; Liew, K.M. Numerical analysis of generalized regularized long wave equation using the element-free kp-Ritz method. Appl. Math.
**2014**, 240, 91–101. [Google Scholar] [CrossRef] - Shen, Q. Local RBF-based differential quadrature collocation method for the boundary layer problems. Eng. Anal. Bound. Elem.
**2010**, 34, 213–228. [Google Scholar] [CrossRef] - Ahsan, M.; Ahmad, I.; Ahmad, M.; Hussian, A. A numerical Haar wavelet-finite difference hybrid method for linear and non-linear Schrödinger equation. Math. Comput. Simul.
**2019**. [Google Scholar] [CrossRef] - Thounthong, P.; Khan, M.; Hussain, I.; Ahmad, I.; Kumam, P. Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations. Mathematics
**2018**, 6, 327. [Google Scholar] [CrossRef] - Kudela, H.; Malecha, Z.M. Eruption of a boundary layer induced by a 2D vortex patch. Fluid Dyn. Res.
**2009**, 41, 055502. [Google Scholar] [CrossRef] - Seydaoğlu, M. A Meshless Method for Burgers’ Equation Using Multiquadric Radial Basis Functions with a Lie-Group Integrator. Mathematics
**2019**, 7, 113. [Google Scholar] [CrossRef] - Shen, Q. Numerical solution of the Sturm-Liouville problem with local RBF-based differential quadrature collocation method. Int. J. Comput. Math.
**2011**, 88, 285–295. [Google Scholar] [CrossRef] - Siraj-ul-Islam; Ahmad, I. A comparative analysis of local meshless formulation for multi-asset option models. Eng. Anal. Bound. Elem.
**2016**, 65, 159–176. [Google Scholar] - Siraj-ul-Islam; Ahmad, I. Local meshless method for PDEs arising from models of wound healing. Appl. Math. Model.
**2017**, 48, 688–710. [Google Scholar] - Ahmad, I.; Siraj-ul-Islam; Khaliq, A.Q.M. Local RBF method for multi-dimensional partial differential equations. Comput. Math. Appl.
**2017**, 74, 292–324. [Google Scholar] [CrossRef] - 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. [Google Scholar] [CrossRef] - Ahmad, I.; Ahsan, M.; Zaheer-ud-Din; Ahmad, M.; Kumam, P. An Efficient Local Formulation for Time-Dependent PDEs. Mathematics
**2019**, 7, 216. [Google Scholar] [CrossRef] - Ahmad, I. Local Meshless Collocation Method for Numerical Solution of Partial Differential Equations. Ph.D. Thesis, University of Engineering and Technology, Peshawar, Pakistan, 2017. [Google Scholar]
- Hussain, A.; ul Haq, S.; Uddin, M. Numerical solution of Klein–Gordon and sine-Gordon equations by meshless method of lines. Eng. Anal. Bound. Elem.
**2013**, 37, 1351–1366. [Google Scholar] [CrossRef] - Khuri, S.; Sayfy, A. A spline collocation approach for the numerical solution of a generalized nonlinear Klein–Gordon equation. Appl. Math. Comput.
**2010**, 216, 1047–1056. [Google Scholar] [CrossRef]

**Figure 2.**Schematic of the local supported domain in 2D geometry for ${n}_{i}=5$ in the upwind technique [34].

**Figure 4.**c versus the ${L}_{\infty}$ error norm of the local meshless method (LMM) (

**left**) and the global meshless method (GMM) (

**right**) for Test Problem 1.

**Figure 7.**c versus the ${L}_{\infty}$ error norm of the LMM (

**left**) and GMM (

**right**) for Test Problem 2.

**Figure 8.**Convergence in space (

**left**) at $t=1$ and convergence in time (

**right**) at $t=0.1$ for the test problem 2.

**Figure 10.**c versus the ${L}_{\infty}$ error norm of the LMM (

**left**) and GMM (

**right**) for Test Problem 3.

**Figure 11.**c versus $\kappa $ (

**left**) and the number of nodal points N versus $\kappa $ (

**right**) of the LMM for Test Problem 3.

**Figure 12.**Numerical solution of the 2D coupled Burgers’ with $Re=300$, $t=1$, $N=61\times 61$ for Test Problem 4.

**Figure 14.**Numerical solution of the 2D coupled Burgers’ combined with the upwind technique for $Re=500$, $t=1$ for Test Problem 4.

**Figure 15.**Numerical solution of the 2D coupled Burgers’ combined with the upwind technique for $Re=1000$, $t=1$ for Test Problem 4.

**Figure 16.**Comparison of the CPU time of the LMM and the GMM with $\Delta t=0.0001$, $t=0.5$ for Test Problem 4.

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

$\mathit{t}$ | FEDF | MQ-RK4 [38] | MQ-Störmer [38] |

0.1 | 4.2548 × ${10}^{-6}$ | 1.90058 × ${10}^{-6}$ | 2.83297 × ${10}^{-5}$ |

0.5 | 3.1827 × ${10}^{-5}$ | 8.16948 × ${10}^{-5}$ | 8.60692 × ${10}^{-5}$ |

1 | 8.8951 × ${10}^{-5}$ | 3.18899 × ${10}^{-4}$ | 3.94758 × ${10}^{-5}$ |

**Table 2.**Spatial convergence rate of the ${L}_{\infty}$ and ${L}_{2}$ error norms for Test Problem 1.

N | $\mathit{\kappa}$ | ${\mathit{L}}_{\mathbf{\infty}}$ | ${\mathit{L}}_{\mathbf{\infty}}$-Rate | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{2}$-Rate |
---|---|---|---|---|---|

11 | 4.5015 × ${10}^{8}$ | 8.8951 × ${10}^{-5}$ | … | 6.4634 × ${10}^{-5}$ | … |

21 | 7.2006 × ${10}^{9}$ | 1.9551 × ${10}^{-5}$ | 2.1858 | 1.3906 × ${10}^{-5}$ | 2.2166 |

31 | 3.6451 × ${10}^{10}$ | 6.9794 × ${10}^{-6}$ | 2.5342 | 4.9459 × ${10}^{-6}$ | 2.5433 |

41 | 1.1520 × ${10}^{11}$ | 3.2008 × ${10}^{-6}$ | 2.7193 | 2.2620 × ${10}^{-6}$ | 2.7288 |

$\mathbf{\Delta}\mathit{t}$ | ${\mathit{L}}_{\mathbf{\infty}}$ | ${\mathit{L}}_{\mathbf{\infty}}$-Rate | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{2}$-Rate |
---|---|---|---|---|

0.1 | 1.9375 × ${10}^{-2}$ | … | 9.7636 × ${10}^{-3}$ | … |

0.05 | 5.7309 × ${10}^{-3}$ | 1.7574 | 3.1601 × ${10}^{-3}$ | 1.6274 |

0.01 | 1.0545 × ${10}^{-3}$ | 1.0518 | 5.7200 × ${10}^{-4}$ | 1.0620 |

0.005 | 5.1010 × ${10}^{-4}$ | 1.0478 | 2.6290 × ${10}^{-4}$ | 1.1215 |

Method | $\mathit{t}=0.01$ | $\mathit{t}=0.02$ | $\mathit{t}=0.1$ | $\mathit{t}=0.5$ | $\mathit{t}=1$ |
---|---|---|---|---|---|

FEDF | 3.1377 × ${10}^{-10}$ | 2.4101 × ${10}^{-9}$ | 2.9157 × ${10}^{-7}$ | 3.3156 × ${10}^{-5}$ | 2.2616 × ${10}^{-4}$ |

MQ-Störmer [38] | 1.57022 × ${10}^{-7}$ | 6.29035 × ${10}^{-7}$ | 1.50205 × ${10}^{-5}$ | 8.84373 × ${10}^{-5}$ | 5.19689 × ${10}^{-5}$ |

GA-Störmer [38] | 5.45453 × ${10}^{-7}$ | 3.31884 × ${10}^{-6}$ | 2.91691 × ${10}^{-4}$ | 1.06862 × ${10}^{-2}$ | 2.33701 × ${10}^{-2}$ |

[39] | 1.7 × ${10}^{-7}$ | 8.4 × ${10}^{-7}$ | 5.4 × ${10}^{-5}$ | 1.2 × ${10}^{-3}$ | 4.3 × ${10}^{-3}$ |

**Table 5.**Spatial convergence rate of the ${L}_{\infty}$ and ${L}_{2}$ error norms for Test Problem 2.

N | $\mathit{\kappa}$ | ${\mathit{L}}_{\mathbf{\infty}}$ | ${\mathit{L}}_{\mathbf{\infty}}$-Rate | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{2}$-Rate |
---|---|---|---|---|---|

11 | 7.3976 × ${10}^{7}$ | 2.2616 × ${10}^{-4}$ | … | 1.9812 × ${10}^{-4}$ | … |

21 | 1.1829 × ${10}^{9}$ | 5.6922 × ${10}^{-5}$ | 1.9903 | 4.9730 × ${10}^{-5}$ | 1.9942 |

31 | 5.9877 × ${10}^{9}$ | 2.5382 × ${10}^{-5}$ | 1.9870 | 2.2114 × ${10}^{-5}$ | 1.9938 |

41 | 1.8923 × ${10}^{10}$ | 1.4326 × ${10}^{-5}$ | 1.9951 | 1.2480 × ${10}^{-5}$ | 1.9955 |

$\mathbf{\Delta}\mathit{t}$ | ${\mathit{L}}_{\mathbf{\infty}}$ | ${\mathit{L}}_{\mathbf{\infty}}$-Rate | ${\mathit{L}}_{2}$ | ${\mathit{L}}_{2}$-Rate |
---|---|---|---|---|

0.1 | 1.6453 × ${10}^{-4}$ | … | 1.4005 × ${10}^{-4}$ | … |

0.05 | 4.1397 × ${10}^{-5}$ | 1.9908 | 3.5238 × ${10}^{-5}$ | 1.9908 |

0.01 | 2.0933 × ${10}^{-6}$ | 1.8543 | 1.6822 × ${10}^{-6}$ | 1.8901 |

0.005 | 7.4903 × ${10}^{-7}$ | 1.4827 | 6.1088 × ${10}^{-7}$ | 1.4614 |

t | 1 | 3 | 5 | 7 | 10 |
---|---|---|---|---|---|

FEDF, $N=11$ | |||||

${L}_{\infty}$ | 3.3468 × ${10}^{-6}$ | 6.1312 × ${10}^{-6}$ | 6.8303 × ${10}^{-6}$ | 6.3888 × ${10}^{-6}$ | 5.1932 × ${10}^{-6}$ |

${L}_{2}$ | 2.1690 × ${10}^{-6}$ | 4.2790 × ${10}^{-6}$ | 5.4027 × ${10}^{-6}$ | 5.1245 × ${10}^{-6}$ | 4.4432 × ${10}^{-6}$ |

Lattice Boltzmann method [10], $N=100$ | |||||

${L}_{\infty}$ | 1.9558 × ${10}^{-3}$ | 1.3664 × ${10}^{-3}$ | 1.5260 × ${10}^{-3}$ | 1.6201 × ${10}^{-3}$ | 1.0465 × ${10}^{-3}$ |

${L}_{2}$ | 1.1135 × ${10}^{-3}$ | 7.6676 × ${10}^{-3}$ | 8.5602 × ${10}^{-3}$ | 9.5926 × ${10}^{-3}$ | 6.9848 × ${10}^{-3}$ |

TPSRBFs method [5], $N=100$ | |||||

${L}_{\infty}$ | 1.2540 × ${10}^{-5}$ | 1.5554 × ${10}^{-5}$ | 3.3792 × ${10}^{-5}$ | 3.7753 × ${10}^{-5}$ | … |

${L}_{2}$ | 6.5422 × ${10}^{-5}$ | 1.1717 × ${10}^{-4}$ | 2.2011 × ${10}^{-4}$ | 2.5892 × ${10}^{-4}$ | … |

MQ quasi-interpolation scheme [11], $N=10$ | |||||

${L}_{\infty}$ | 1.25905 | 1.5428 × ${10}^{-5}$ | 3.3625 × ${10}^{-5}$ | 3.7412 × ${10}^{-5}$ | … |

${L}_{2}$ | 2.0694 × ${10}^{-5}$ | 3.7065 × ${10}^{-5}$ | 6.9684 × ${10}^{-5}$ | 8.1943 × ${10}^{-5}$ | … |

**Table 8.**Comparison of the numerical results of the 2D coupled Burgers’ equation for Test Problem 4.

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

N/$Re$ | 1 | 20 | 100 | 1 | 20 | 100 | ||

121 | FEDF | 1.3 × ${10}^{-8}$ | 8.5 × ${10}^{-5}$ | 7.3 × ${10}^{-3}$ | 3.1 × ${10}^{-8}$ | 8.5 × ${10}^{-5}$ | 7.3 × ${10}^{-3}$ | |

LMAPS [15] | 7.4 × ${10}^{-6}$ | 9.7 × ${10}^{-5}$ | 7.4 × ${10}^{-3}$ | 1.0 × ${10}^{-5}$ | 8.8 × ${10}^{-5}$ | 7.4 × ${10}^{-3}$ | ||

LDQ [15] | 1.4 × ${10}^{-5}$ | 6.7 × ${10}^{-5}$ | 7.3 × ${10}^{-3}$ | 1.9 × ${10}^{-5}$ | 7.5 × ${10}^{-5}$ | 7.3 × ${10}^{-3}$ | ||

441 | FEDF | 3.4 × ${10}^{-9}$ | 2.1 × ${10}^{-5}$ | 2.0 × ${10}^{-3}$ | 8.0 × ${10}^{-9}$ | 2.1 × ${10}^{-5}$ | 2.0 × ${10}^{-3}$ | |

LMAPS [15] | 1.9 × ${10}^{-6}$ | 2.4 × ${10}^{-5}$ | 2.1 × ${10}^{-3}$ | 2.6 × ${10}^{-6}$ | 2.1 × ${10}^{-5}$ | 2.0 × ${10}^{-3}$ | ||

LDQ [15] | 3.4 × ${10}^{-6}$ | 1.7 × ${10}^{-5}$ | 2.0 × ${10}^{-3}$ | 4.7 × ${10}^{-6}$ | 1.9 × ${10}^{-5}$ | 2.0 × ${10}^{-3}$ | ||

961 | FEDF | 5.6 × ${10}^{-9}$ | 9.5 × ${10}^{-6}$ | 8.4 × ${10}^{-4}$ | 6.4 × ${10}^{-9}$ | 9.3 × ${10}^{-6}$ | 8.4 × ${10}^{-4}$ | |

LMAPS [15] | 8.3 × ${10}^{-7}$ | 1.1 × ${10}^{-5}$ | 8.5 × ${10}^{-4}$ | 1.1 × ${10}^{-6}$ | 9.2 × ${10}^{-6}$ | 8.4 × ${10}^{-4}$ | ||

LDQ [15] | 1.5 × ${10}^{-6}$ | 7.5 × ${10}^{-6}$ | 8.4 × ${10}^{-4}$ | 2.1 × ${10}^{-6}$ | 8.1 × ${10}^{-6}$ | 8.2 × ${10}^{-4}$ |

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

$(x,y)$ | $(0.1,0.1)$ | $(0.3,0.3)$ | $(0.5,0.5)$ | $(0.1,0.1)$ | $(0.3,0.3)$ | $(0.5,0.5)$ | |

Exact | 0.500482 | 0.500482 | 0.500482 | 0.999518 | 0.999518 | 0.999518 | |

FEDF | 0.500470 | 0.500443 | 0.500417 | 0.999530 | 0.999557 | 0.999583 | |

Global RBFs method [14] | 0.50035 | 0.50042 | 0.50046 | 0.99936 | 0.99951 | 0.99958 | |

Eulerian-Lagrangian method [13] | 0.50012 | 0.50042 | 0.50041 | 0.99946 | 0.99938 | 0.99941 | |

Finite difference method [12] | 0.49983 | 0.49977 | 0.49973 | 0.99826 | 0.99861 | 0.99821 |

$\mathit{t}=1$ | $\mathit{t}=5$ | $\mathit{t}=10$ | $\mathit{t}=15$ | $\mathit{t}=20$ |
---|---|---|---|---|

${L}_{\infty}$ | ||||

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

CPU time (in seconds) | ||||

2.54 | 2.59 | 2.71 | 2.85 | 2.88 |

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**MDPI and ACS Style**

Ahmad, I.; Ahsan, M.; Hussain, I.; Kumam, P.; Kumam, W.
Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method. *Symmetry* **2019**, *11*, 394.
https://doi.org/10.3390/sym11030394

**AMA Style**

Ahmad I, Ahsan M, Hussain I, Kumam P, Kumam W.
Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method. *Symmetry*. 2019; 11(3):394.
https://doi.org/10.3390/sym11030394

**Chicago/Turabian Style**

Ahmad, Imtiaz, Muhammad Ahsan, Iltaf Hussain, Poom Kumam, and Wiyada Kumam.
2019. "Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method" *Symmetry* 11, no. 3: 394.
https://doi.org/10.3390/sym11030394