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Two stages six-step method with eliminated phase-lag and its first, second, third and fourth derivatives for the approximation of the Schrödinger equation

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Abstract

For the first time in the literature, a new two-stages symmetric six-step algorithm is developed and analyzed. The new algorithm has:

  • tenth algebraic order (which is the highest possible order),

  • vanished phase-lag and its first, second, third and fourth derivatives.

  • good stability properties i.e. an interval of periodicity, equal to \(\left( 0, 133.36 \right) \),

  • the approximation of the first stage of the algorithm is done on the point \(x_{n+3}\) and no at the usual point \(x_{n}\).

We also present a full analysis of the new algorithm (i.e. error, stability and interval of periodicity analysis). Finally, we also examine the effectiveness of the new obtained algorithm by comparing it with well known algorithms and very recently produced algorithms in the literature. Three stages of comparison for the efficiency of the algorithm are used:

  • Comparison on local truncation error analysis,

  • Comparison on stability analysis,

  • Comparison on accuracy and computational effectiveness of the solution of the Schrödinger equation.

The theoretical and numerical achievements lead to the conclusion that the new algorithm is more efficient than other well known or recently obtained algorithms.

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Authors and Affiliations

  1. Group of Modern Computational Methods, Ural Federal University, 19 Mira Street, Yekaterinburg, Russian Federation, 620002

    Maxim A. Medvedev & T. E. Simos

  2. Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, 221 00, Tripoli, Greece

    T. E. Simos

  3. 10 Konitsis Street, Amphithea - Paleon Phaleron, 175 64, Athens, Greece

    T. E. Simos

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  1. Maxim A. Medvedev
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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.

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Medvedev, M.A., Simos, T.E. Two stages six-step method with eliminated phase-lag and its first, second, third and fourth derivatives for the approximation of the Schrödinger equation. J Math Chem 55, 961–986 (2017). https://doi.org/10.1007/s10910-016-0711-y

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