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

机译：基于微分求积法的非线性Klein-Gordon方程数值模拟方案

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摘要

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.
机译：目的-本文的目的是提出一种基于正向有限差分，拟线性化过程和多项式微分求积法的数值方案，以找到具有Dirichlet和Neumann边界条件的非线性Klein-Gordon方程的数值解。设计/方法/方法-第一步，通过前向差分法离散时间导数。然后，使用准线性化过程来解决方程中的非线性问题。最后，采用微分求积法（DQM）进行完全离散化，可以得到一个线性方程组，该系统可以通过高斯消除法求解。结果-通过几个测试示例证明了所提出方法的准确性。发现数值结果与精确解非常吻合，并且文献中存在数值解。所提出的方案可以扩展到多维问题。创意/价值-本方案的主要优点是，该方案通过选择较少的网格点数，可以为精确解决方案提供非常准确和相似的结果。其次，通过选择较少的网格点数和较大的时间步长，该方案比（Dehghan和Shokri，2009； Pekmen和Tezer-Sezgin，2012）具有更高的准确性。而且，该方案可以扩展到多维问题。

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