New Runge–Kutta type symmetric twostep 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 twostep method with the following properties: (1) the new scheme is of symmetric type, (2) the new scheme is of twostep, (3) the new scheme is of fivestages, (4) the new scheme is of twelfthalgebraic order, (5) the new scheme has eliminated the phaselag 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 Pstable [with interval of periodicity equal to \(\left( 0, \infty \right) \)] and (8) the new scheme builded based on the following approximations: 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.

the first stage is approximation on the point \(x_{n1}\),

the second stage is approximation on the point \(x_{n1}\),

the third stage is approximation on the point \(x_{n1}\),

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

the fifth stage is approximation 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
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)
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