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New hybrid symmetric two step scheme with optimized characteristics for second order problems

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Abstract

In the present paper, for the first time in the literature, we build a new three-stages symmetric two-step finite difference pair with optimized properties. In more details the new method: (1) is of symmetric type, (2) is of two-step algorithm, (3) is of three-stages—i.e. hybrid or Runge–Kutta type, (4) it is of tenth-algebraic order, (5) it has vanished the phase-lag and its first and second derivatives, (6) it has optimized stability properties for the general problems, (7) it is a P-stable finite difference scheme since it has an interval of periodicity equal to \(\left( 0, \infty \right) \). The new Runge–Kutta type algorithm is builded based on the following approximations:

  • An approximation determined on the first layer on the point \(x_{n-1}\),

  • An approximation determined on the second layer on the point \(x_{n}\) and finally,

  • An approximation determined on the third (final) layer on the point \(x_{n+1}\),

A full theoretical analysis (local truncation error analysis, comparative error analysis and stability and interval of periodicity analysis) is given for the new builded finite difference pair. The effectiveness of the new builded hybrid scheme is evaluated on the numerical solution of systems of coupled differential equations of the Schrödinger type.

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Acknowledgements

The research was funded by a grant of the Russian Foundation for Basic Research (RFBR) for the Project No. 16-38-60114-mol-a-dk.

Author information

Authors and Affiliations

  1. Group of Numerical and Applied Mathematics on Urgent Problems of Energy and Power Engineering, Faculty of Power Engineering, Ulyanovsk State Technical University, Severny Venets Street 32, Ulyanovsk, Russia, 432027

    V. N. Kovalnogov & R. V. Fedorov

  2. Dr. T.E. Simos, 10 Konitsis Street, Amfithea - Paleon Faliron, 175 64, Athens, Greece

    T. E. Simos

  3. Department of Mathematics, College of Sciences, King Saud University, P. O. Box 2455, Riyadh, 11451, Kingdom of Saudi Arabia

    T. E. Simos

  4. Group of Modern Computational Methods, Ural Federal University, 19 Mira Street, Yekaterinburg, Russia, 620002

    T. E. Simos

  5. Department of Automation Engineering, TEI of Sterea Hellas, Psachna Campus, 344 00, Psachna, Greece

    T. E. Simos

  6. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece

    T. E. Simos

Authors
  1. V. N. Kovalnogov
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  2. R. V. Fedorov
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  3. T. E. Simos
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Corresponding author

Correspondence to T. E. Simos.

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T.E. Simos: Highly Cited Researcher (http://isihighlycited.com/), Active Member of the European Academy of Sciences and Arts. Active Member of the European Academy of Sciences. Corresponding Member of European Academy of Arts, Sciences and Humanities.

Appendix

Appendix

where \(\psi \left( x \right) = \psi _{n}\).

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Kovalnogov, V.N., Fedorov, R.V. & Simos, T.E. New hybrid symmetric two step scheme with optimized characteristics for second order problems. J Math Chem 56, 2816–2844 (2018). https://doi.org/10.1007/s10910-018-0920-7

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