Numerical Simulation of PDEs by Local Meshless Differential Quadrature Collocation Method

Imtiaz Ahmad,Poom Kumam,Muhammad Ahsan,Iltaf Hussain,Wiyada Kumam

doi:10.3390/sym11030394

Abstract

In this paper, a local meshless differential quadrature collocation method based on radial basis functions is proposed for the numerical simulation of one-dimensional Klein–Gordon, two-dimensional coupled Burgers’, and regularized long wave equations. Both local and global meshless collocation procedures are used for spatial discretization, which convert the mentioned partial differential equations into a system of ordinary differential equations. The obtained system has been solved by the forward Euler difference formula. An upwind technique is utilized in the case of the convection-dominated coupled Burgers’ model equation. Having no need for the mesh in the problem domain and being less sensitive to the variation of the shape parameter as compared to global meshless methods are the salient features of the local meshless method. Both rectangular and non-rectangular domains with uniform and scattered nodal points are considered. Accuracy, efficacy, and the ease of implementation of the proposed method are shown via test problems.

Highlights

The Klein–Gordon (KG) equation can describe various vital phenomena in chemical and physical sciences
It is noted that the global meshless method (GMM), which is based on the global interpolation paradigm, has faced the problems of dense ill-conditioned matrices and finding the optimum value of the shape parameter
The proposed local meshless differential quadrature collocation method (LMM) is combined with a technique based on the local supported domain, called an upwind technique, in the case of convection-dominated PDE models

Summary

Introduction

The Klein–Gordon (KG) equation can describe various vital phenomena in chemical and physical sciences. The 2D nonlinear regularized long wave (RLW) equation with initial and boundary conditions can be written as [16,17],. The regularized long wave (RLW) model equation has described many physical phenomena [18]. It is noted that the global meshless method (GMM), which is based on the global interpolation paradigm, has faced the problems of dense ill-conditioned matrices and finding the optimum value of the shape parameter. The local meshless differential quadrature collocation method (LMM) based on radial basis functions (RBFs) is proposed for the numerical simulation of 1D nonlinear KG, 2D coupled.

Implementation of the Numerical Method

Implementation of LMM for the KG Equation

Implementation of LMM for the 2D Model Equations

Numerical Analysis

Conclusions