Journal of Mathematical Chemistry

, Volume 56, Issue 8, pp 2302–2340

# New hybrid two-step method with optimized phase and stability characteristics

Original Paper

## Abstract

In this paper and for the first time in the literature, we develop a new Runge–Kutta type symmetric two-step finite difference pair with the following characteristics:
• the new algorithm is of symmetric type,

• the new algorithm is of two-step,

• the new algorithm is of five-stages,

• the new algorithm is of twelfth-algebraic order,

• the new algorithm is based on the following approximations:
1. 1.

the first layer on the point $$x_{n-1}$$,

2. 2.

the second layer on the point $$x_{n-1}$$,

3. 3.

the third layer on the point $$x_{n-1}$$,

4. 4.

the fourth layer on the point $$x_{n}$$ and finally,

5. 5.

the fifth (final) layer on the point $$x_{n+1}$$,

• the new algorithm has vanished the phase-lag and its first, second, third and fourth derivatives,

• the new algorithm has improved stability characteristics for the general problems,

• the new algorithm is of P-stable type since it has an interval of periodicity equal to $$\left( 0, \infty \right)$$.

For the new developed algorithm we present a detailed numerical analysis (local truncation error and stability analysis). The effectiveness of the new developed algorithm is evaluated with the approximate solution of coupled differential equations arising from the Schrödinger type.

## Keywords

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

65L05

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## Authors and Affiliations

• V. N. Kovalnogov
• 1
• R. V. Fedorov
• 1
• A. A. Bondarenko
• 2
• T. E. Simos
• 3
• 4
• 5
• 6
• 7
Email author
1. 1.Group of Numerical and Applied Mathematics on Urgent Problems of Energy and Power Engineering, Faculty of Power EngineeringUlyanovsk State Technical UniversityUlyanovskRussian Federation
2. 2.Ulyanovsk Institute of Civil Aviation named after Chief Marshal of Aviation BP Bugaev State Technical UniversityUlyanovskRussian Federation
3. 3.Department of Mathematics, College of SciencesKing Saud UniversityRiyadhSaudi Arabia
4. 4.Group of Modern Computational MethodsUral Federal UniversityEkaterinburgRussian Federation
5. 5.Department of Automation EngineeringTEI of Sterea HellasPsachnaGreece
6. 6.Section of Mathematics, Department of Civil EngineeringDemocritus University of ThraceXanthiGreece
7. 7.AthensGreece