A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation

Abstract

Purpose

The purpose of this paper is to propose a numerical scheme based on forward finite difference, quasi-linearisation process and polynomial differential quadrature method to find the numerical solutions of nonlinear Klein-Gordon equation with Dirichlet and Neumann boundary condition.

Design/methodology/approach

In first step, time derivative is discretised by forward difference method. Then, quasi-linearisation process is used to tackle the non-linearity in the equation. Finally, fully discretisation by differential quadrature method (DQM) leads to a system of linear equations which is solved by Gauss-elimination method.

Findings

The accuracy of the proposed method is demonstrated by several test examples. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions exist in literature. The proposed scheme can be expended for multidimensional problems.

Originality/value

The main advantage of the present scheme is that the scheme gives very accurate and similar results to the exact solutions by choosing less number of grid points. Secondly, the scheme gives better accuracy than (Dehghan and Shokri, 2009; Pekmen and Tezer-Sezgin, 2012) by choosing less number of grid points and big time step length. Also, the scheme can be extended for multidimensional problems.

Citation

Verma, A., Jiwari, R. and Kumar, S. (2014), "A numerical scheme based on differential quadrature method for numerical simulation of nonlinear Klein-Gordon equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 24 No. 7, pp. 1390-1404. https://doi.org/10.1108/HFF-01-2013-0014