New hybrid symmetric two step scheme with optimized characteristics for second order problems
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Abstract
In the present paper, for the first time in the literature, we build a new threestages symmetric twostep finite difference pair with optimized properties. In more details the new method: (1) is of symmetric type, (2) is of twostep algorithm, (3) is of threestages—i.e. hybrid or Runge–Kutta type, (4) it is of tenthalgebraic order, (5) it has vanished the phaselag and its first and second derivatives, (6) it has optimized stability properties for the general problems, (7) it is a Pstable finite difference scheme since it has an interval of periodicity equal to \(\left( 0, \infty \right) \). The new Runge–Kutta type algorithm is builded based on the following approximations: A full theoretical analysis (local truncation error analysis, comparative error analysis and stability and interval of periodicity analysis) is given for the new builded finite difference pair. The effectiveness of the new builded hybrid scheme is evaluated on the numerical solution of systems of coupled differential equations of the Schrödinger type.

An approximation determined on the first layer on the point \(x_{n1}\),

An approximation determined on the second layer on the point \(x_{n}\) and finally,

An approximation determined on the third (final) layer on the point \(x_{n+1}\),
Keywords
Phaselag Derivative of the phaselag Initial value problems Oscillating solution Symmetric Hybrid Multistep Schrödinger equationMathematics Subject Classification
65L05Notes
Acknowledgements
The research was funded by a grant of the Russian Foundation for Basic Research (RFBR) for the Project No. 163860114moladk.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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