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Padé numerical schemes for the sine-Gordon equation

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  • Martin-Vergara, Francisca
  • Rus, Francisco
  • Villatoro, Francisco R.

Abstract

The sine-Gordon equation turn up in several problems in science and engineering. Although it is integrable, in practical applications, its numerical solution is powerful and versatile. Four novel implicit finite difference methods based on (q, s) Padé approximations with (q+s)th order in space have been developed and analyzed for this equation; all share the same treatment for the nonlinearity and integration in time. Concretely, (0,4), (2,2), (2,4), and (4,4) Padé methods; additionally, the energy conserving, Strauss–Vázquez scheme has been considered in a (0,2) Padé implementation. These methods have been compared among them for both the kink–antikink and breather solutions in terms of global error, computational cost and energy conservation. The (0,4) and (2,4) Padé methods are the most cost-effective ones for small and large global error, respectively. Our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations.

Suggested Citation

  • Martin-Vergara, Francisca & Rus, Francisco & Villatoro, Francisco R., 2019. "Padé numerical schemes for the sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 232-243.
    • Handle: RePEc:eee:apmaco:v:358:y:2019:i:c:p:232-243
    DOI: 10.1016/j.amc.2019.04.042
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    References listed on IDEAS

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    1. Jiang, Chaolong & Sun, Jianqiang & Li, Haochen & Wang, Yifan, 2017. "A fourth-order AVF method for the numerical integration of sine-Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 144-158.
    2. Brugnano, L. & Frasca Caccia, G. & Iavernaro, F., 2015. "Energy conservation issues in the numerical solution of the semilinear wave equation," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 842-870.
    3. Dehghan, Mehdi & Shokri, Ali, 2008. "A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 700-715.
    4. Nemati Saray, Behzad & Lakestani, Mehrdad & Cattani, Carlo, 2018. "Evaluation of mixed Crank–Nicolson scheme and Tau method for the solution of Klein–Gordon equation," Applied Mathematics and Computation, Elsevier, vol. 331(C), pages 169-181.
    5. Bratsos, A.G., 2007. "A third order numerical scheme for the two-dimensional sine-Gordon equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 76(4), pages 271-282.
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    Cited by:

    1. Martin-Vergara, Francisca & Rus, Francisco & Villatoro, Francisco R., 2021. "Fractal structure of the soliton scattering for the graphene superlattice equation," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    2. Martin-Vergara, Francisca & Rus, Francisco & Villatoro, Francisco R., 2022. "Numerical search for the stationary quasi-breather of the graphene superlattice equation," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).

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