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A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation

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Abstract

A Runge–Kutta type (four stages) eighth algebraic order two-step method with phase-lag and its first, second, third and fourth derivatives equal to zero is produced in this paper. We also study the results of elimination of the phase-lag and its derivatives on the efficiency of the method. Our studies consist: (1) the construction of the method, (2) the determination of the local truncation error of the proposed method, (3) the investigation of the local truncation error analysis using the comparison with other similar methods of the literature, (4) the computation of the interval of periodicity (stability interval) of the developed method. For this calculation we use a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, (5) the definition of the error estimation based on methods with different algebraic orders and (6) the investigation of the effectiveness of the new obtained method studying the numerical solution of the coupled differential equations arising from the Schrödinger equation.

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Notes

  1. where \(S\) is a set of distinct points

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Author information

Authors and Affiliations

  1. School of Information Engineering, Chang’an University, Xi’an, 710064, China

    Kenan Mu

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

    T. E. Simos

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

    T. E. Simos

  4. 10 Konitsis Street, Amfithea - Paleon Faliron, 175 64, Athens, Greece

    T. E. Simos

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  1. Kenan Mu
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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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Mu, K., Simos, T.E. A Runge–Kutta type implicit high algebraic order two-step method with vanished phase-lag and its first, second, third and fourth derivatives for the numerical solution of coupled differential equations arising from the Schrödinger equation. J Math Chem 53, 1239–1256 (2015). https://doi.org/10.1007/s10910-015-0484-8

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