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Z. Naturforsch. 66a, 735 – 744
(2011)
doi:10.5560/ZNA.2011-0038
Fast and High Accuracy Numerical Methods for the Solution of Nonlinear Klein–Gordon Equations
1 Department of Mathematics, Faculty of Science, University of Kashan, Kashan, Iran
2 Department of Mathematics, Faculty of Science, Alzahra University, Tehran, Iran
Received April 13, 2011 / revised July 6, 2011
Reprint requests to: A. M.; E-mail:
akbar_mohebbi@aut.ac.ir
In this work we propose fast and high accuracy numerical methods for the solution of the one-dimensional nonlinear Klein–Gordon (KG) equations. These methods are based on applying fourth-order time-stepping schemes in combination with discrete Fourier transform to numerically solve the KG equations. After transforming each equation to a system of ordinary differential equations, the linear operator is not diagonal, but we can implement the methods such as for the diagonal case which reduces the time in the central processing unit (CPU). In addition, the conservation of energy in KG equations is investigated. Numerical results obtained from solving several problems possessing periodic, single, and breather-soliton waves show the high efficiency and accuracy of the mentioned methods.
Key words: Klein–Gordon Equation; Exponential Time Differencing; Integrating Factor; Spectral Methods; High Accuracy; Soliton; Conservation of Energy.