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New Runge–Kutta type symmetric two-step method with optimized characteristics

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Abstract

In this paper and for the first time in the literature, we build a new hybrid symmetric two-step method with the following properties: (1) the new scheme is of symmetric type, (2) the new scheme is of two-step, (3) the new scheme is of five-stages, (4) the new scheme is of twelfth-algebraic order, (5) the new scheme has eliminated the phase-lag and its first, second, third, fourth and fifth derivatives, (6) the new scheme has improved stability characteristics for the general problems, (7) the new scheme is P-stable [with interval of periodicity equal to \(\left( 0, \infty \right) \)] and (8) the new scheme builded based on the following approximations:

  • the first stage is approximation on the point \(x_{n-1}\),

  • the second stage is approximation on the point \(x_{n-1}\),

  • the third stage is approximation on the point \(x_{n-1}\),

  • the fourth stage is approximation on the point \(x_{n}\) and finally,

  • the fifth stage is approximation on the point \(x_{n+1}\),

For the new builded scheme we give a full numerical analysis (local truncation error and stability analysis). The efficiency of the new builded scheme is examined with the numerical solution of systems of coupled differential equations of the Schrödinger type.

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Authors and Affiliations

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

    Ke Yan

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

    T. E. Simos

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

    T. E. Simos

  4. Department of Automation Engineering, TEI of Sterea Hellas, Psachna Campus, 34400, Psachna, Greece

    T. E. Simos

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

    T. E. Simos

  6. 10 Konitsis Street, Amphithea - Paleon Phaleron, 175 64, Athens, Greece

    T. E. Simos

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  1. Ke Yan
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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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Yan, K., Simos, T.E. New Runge–Kutta type symmetric two-step method with optimized characteristics. J Math Chem 56, 2454–2484 (2018). https://doi.org/10.1007/s10910-018-0899-0

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