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Journal of Mathematical Chemistry

, Volume 56, Issue 4, pp 982–1010 | Cite as

New five-stages finite difference pair with optimized phase properties

  • Xin Shi
  • T. E. SimosEmail author
Original Paper

Abstract

A new five-stages symmetric two-step finite difference pair is developed, for the first time in the literature, in this paper. This new finite difference pair has optimal phase and stability properties The main characteristics of the new finite difference pair are:
  1. 1.

    it is of symmetric type,

     
  2. 2.

    it is of two-step algorithm,

     
  3. 3.

    it is of five-stages,

     
  4. 4.

    it is of twelfth-algebraic order,

     
  5. 5.
    the new nonlinear finite difference pair is produced using the following approximations:
    • An approximation developed on the first layer on the point \(x_{n-1}\),

    • An approximation developed on the second layer on the point \(x_{n-1}\),

    • An approximation developed on the third layer on the point \(x_{n-1}\),

    • An approximation developed on the fourth layer on the point \(x_{n}\) and finally,

    • An approximation developed on the fifth (final) layer on the point \(x_{n+1}\),

     
  6. 6.

    it has vanished the phase-lag and its first derivative,

     
  7. 7.

    it has optimized stability properties for the general problems,

     
  8. 8.

    it is a P-stable method since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).

     
A full numerical analysis (error and stability analysis) is given for the new finite difference pair.

The effectiveness of the new finite difference pair is evaluated by applying it on the approximate solution of systems of coupled differential equations of the Schrödinger form.

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

Conflicts of interest

The authors declare that they have no conflict of interest.

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Authors and Affiliations

  1. 1.School of Information EngineeringChang’an UniversityXi’anPeople’s Republic of China
  2. 2.Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information ScienceNeijiang Normal UniversityNeijiangPeople’s Republic of China
  3. 3.AthensGreece

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