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A family of embedded explicit six-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation: development and theoretical analysis

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Abstract

In this paper, we introduce for the first time in the literature a family of embedded explicit symmetric six-step methods. The family of the proposed methods has vanishing of the phase-lag and its derivatives. More specifically:

  • the first method of the family is an eight algebraic order method with phase-lag and its first and second derivatives vanished,

  • the second method of the family is an eight algebraic order method with phase-lag and its first, second and third derivatives vanished,

The above new family of the methods is investigated by studying (1) the development of the methods, i.e. the computation of the coefficients of the methods of the family in order the phase-lag and its first, second and third derivatives of the phase-lag to be vanished, (2) the computation of the formulae of the local truncation error of the methods of the new family of methods, (3) the comparative local truncation error analysis which will be taken place by applying the new family of methods on a scalar test problem (which is the time independent radial Schrödinger equation) and (4) the stability analysis of the methods of the new proposed family which will be taken place by applying the new obtained method to a scalar test equation with frequency different than the frequency of the scalar test equation for the phase-lag analysis. Finally, theoretical results are produced on the efficiency of the new developed methods.

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Notes

  1. Where S is a set of distinct points.

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Authors and Affiliations

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

    Ibraheem Alolyan & T. E. Simos

  2. 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

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

    T. E. Simos

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  1. Ibraheem Alolyan
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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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Alolyan, I., Simos, T.E. A family of embedded explicit six-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation: development and theoretical analysis. J Math Chem 54, 1159–1186 (2016). https://doi.org/10.1007/s10910-016-0611-1

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