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A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation

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Abstract

A family of two stage low computational cost symmetric two-step methods with vanished phase-lag and its derivatives is developed in this paper. More specifically we produce:

  • a two-stage symmetric two-step eighth algebraic order method which has the phase-lag and its first, second and third derivatives vanished and

  • a two-stage symmetric two-step sixth algebraic order method, which is P-stable and has the phase-lag and its first and second derivatives vanished.

The local truncation error, the interval of periodicity and the effect of the vanishing of the phase-lag and its derivatives on the efficiency of the obtained method are also studied in this paper.

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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

    Fei Hui

  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. Fei Hui
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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.

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Appendix: Formulae of the derivatives of \(q_{n}\)

Appendix: Formulae of the derivatives of \(q_{n}\)

Formulae of the derivatives which presented in the formulae of the Local Truncation Errors:

$$\begin{aligned} y_{n}^{(2)}= & {} \left( V(x)-V_{c} + G\right) \, y(x) \\ y_{n}^{(3)}= & {} \left( {\frac{d}{dx}}g \left( x \right) \right) y \left( x \right) + \left( g \left( x \right) +G \right) {\frac{d}{dx}}y \left( x \right) \\ y_{n}^{(4)}= & {} \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) y \left( x \right) +2\, \left( {\frac{d}{dx}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) + \left( g \left( x \right) +G \right) ^{2}y \left( x \right) \\ y_{n}^{(5)}= & {} \left( {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \right) y \left( x \right) +3\, \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) +\,4\, \left( g \left( x \right) +G \right) y \left( x \right) {\frac{d}{dx}}g \left( x \right) \\&+\, \left( g \left( x \right) +G \right) ^{2}{\frac{d}{dx}}y \left( x \right) \\ y_{n}^{(6)}= & {} \left( {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) \right) y \left( x \right) +4\, \left( {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) +7\, \left( g \left( x \right) +G \right) y \left( x \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \\&+\,4\, \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}y \left( x \right) +6\, \left( g \left( x \right) +G \right) \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d}{dx}}g \left( x \right) + \left( g \left( x \right) +G \right) ^{3}y \left( x \right) \\ y_{n}^{(7)}= & {} \left( {\frac{d^{5}}{d{x}^{5}}}g \left( x \right) \right) y \left( x \right) +5\, \left( {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) +11\, \left( g \left( x \right) +G \right) y \left( x \right) {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \\&+\,15\, \left( {\frac{d}{dx}}g \left( x \right) \right) y \left( x \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) +13\, \left( g \left( x \right) +G \right) \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \\&+\,10\, \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}{\frac{d}{dx}}y \left( x \right) +9\, \left( g \left( x \right) +G \right) ^{2}y \left( x \right) {\frac{d}{dx}}g \left( x \right) + \left( g \left( x \right) +G \right) ^{3}{\frac{d}{dx}}y \left( x \right) \\ y_{n}^{(8)}= & {} \left( {\frac{d^{6}}{d{x}^{6}}}g \left( x \right) \right) y \left( x \right) \,+\,6\, \left( {\frac{d^{5}}{d{x}^{5}}}g \left( x \right) \right) {\frac{d}{dx}}y \left( x \right) +\,16\, \left( g \left( x \right) +G \right) y \left( x \right) {\frac{d^{4}}{d{x}^{4}}}g \left( x \right) \\&+\,26\, \left( {\frac{d}{dx}}g \left( x \right) \right) y \left( x \right) {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) +24\, \left( g \left( x \right) +G \right) \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d^{3}}{d{x}^{3}}}g \left( x \right) \\&+\,15\, \left( {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \right) ^{2}y \left( x \right) +48\, \left( {\frac{d}{dx}}g \left( x \right) \right) \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) \nonumber \\&+\,22\, \left( g \left( x \right) +G \right) ^{2}y \left( x \right) {\frac{d^{2}}{d{x}^{2}}}g \left( x \right) +28\, \left( g \left( x \right) +G \right) y \left( x \right) \left( {\frac{d}{dx}}g \left( x \right) \right) ^{2}\\&+12\, \left( g \left( x \right) +G \right) ^{2} \left( {\frac{d}{dx}}y \left( x \right) \right) {\frac{d}{dx}}g \left( x \right) +\, \left( g \left( x \right) +G \right) ^{4}y \left( x \right) \ldots \\ \end{aligned}$$

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Hui, F., Simos, T.E. A new family of two stage symmetric two-step methods with vanished phase-lag and its derivatives for the numerical integration of the Schrödinger equation. J Math Chem 53, 2191–2213 (2015). https://doi.org/10.1007/s10910-015-0545-z

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