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.

Keywords

Phase-lag Derivative of the phase-lag Initial value problems Oscillating solution Symmetric Hybrid Multistep Schrödinger equation 

Mathematics Subject Classification

65L05 

Notes

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Supplementary material

10910_2018_899_MOESM1_ESM.pdf (130 kb)
Supplementary material 1 (pdf 130 KB)

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Information EngineeringChang’an UniversityXi’anPeople’s Republic of China
  2. 2.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
  3. 3.Group of Modern Computational MethodsUral Federal UniversityEkaterinburgRussian Federation
  4. 4.Department of Automation EngineeringTEI of Sterea HellasPsachnaGreece
  5. 5.Section of Mathematics, Department of Civil EngineeringDemocritus University of ThraceXanthiGreece
  6. 6.AthensGreece

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