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Fourth-order compact solution of the nonlinear Klein-Gordon equation

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Abstract

In this work we propose a fourth-order compact method for solving the one-dimensional nonlinear Klein-Gordon equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivative and a fourth-order A-stable diagonally-implicit Runge-Kutta-Nyström (DIRKN) method for the time integration of the resulting nonlinear second-order system of ordinary differential equations. The proposed method has fourth order accuracy in both space and time variables and is unconditionally stable. Numerical results obtained from solving several problems possessing periodic, kinks, single and double-soliton waves show that the combination of a compact finite difference approximation of fourth order and a fourth-order A-stable DIRKN method gives an efficient algorithm for solving these problems.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Mehdi Dehghan & Akbar Mohebbi

  2. Department of Mathematics, Faculty of Science, University of Kashan, Kashan, Iran

    Akbar Mohebbi

  3. Department of Mathematics, Faculty of Science, Alzahra University, Tehran, Iran

    Zohreh Asgari

Authors
  1. Mehdi Dehghan
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  2. Akbar Mohebbi
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  3. Zohreh Asgari
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Correspondence to Mehdi Dehghan.

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Dehghan, M., Mohebbi, A. & Asgari, Z. Fourth-order compact solution of the nonlinear Klein-Gordon equation. Numer Algor 52, 523–540 (2009). https://doi.org/10.1007/s11075-009-9296-x

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  • DOI: https://doi.org/10.1007/s11075-009-9296-x

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