IJPAM: Volume 111, No. 2 (2016)
Title
A FOUR-STAGE THIRD-ORDER SYMPLECTIC EXPLICITTRIGONOMETRICALLY-FITTED RUNGE-KUTTA-NYSTRÖM
METHOD FOR THE NUMERICAL INTEGRATION OF
OSCILLATORY INITIAL-VALUE PROBLEMS
Authors
M.A. Demba, N. Senu, F. IsmailDepartment of Mathematics
Universiti Putra Malaysia
43400 UPM Serdang, Selangor, MALAYSIA
Department of Mathematics
and Institute for Mathematical Research
Universiti Putra Malaysia
43400 UPM Serdang, Selangor, MALAYSIA
Abstract
In this work, a third-order four-stage symplectic explicit trigonometrically-fitted Runge-Kutta-Nyström (RKN) method for the numerical integration of second order initial value problems with oscillatory solutions based on Simos technique is constructed. The numerical results obtain signify the accuracy of the proposed method in comparison with other symplectic and non-symplectic RKN methods.History
Received: May 22, 2016
Revised: November 12, 2016
Published: December 11, 2016
AMS Classification, Key Words
AMS Subject Classification: 65L05, 65L06
Key Words and Phrases: trigonometric fitting, RKN methods, periodic initial value problems
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Bibliography
- 1
- H. Van de Vyver,``A symplectic exponentially fitted modified Runge-Kutta-Nyström method for the numerical integration of orbital problems ",New Astronomy, 10 (2005), 261-269.
- 2
- H. Van de Vyver, ``A fourth-order symplectic exponentially-fitted integrator", Computer Physics Communications, 174 (2006), 255-262.
- 3
- H. Van de Vyver, ```A Symplectic Runge-Kutta-Nyström method with minimal phase-lag", Physics Letters A, 260 (2014), 482-493.
- 4
- A. Tocino, and J. Vigo-Aguiar, ``Symplectic condition for exponential fitting Runge-Kutta-Nyström methods", Mathematical and Computer Modelling, 42 (2005), 873-876.
- 5
- Z. Kalogiratou, T. Monovasilis and T. E. Simos, ``A fourth order modified trigonometrically-fitted symplectic Runge-Kutta-Nyström method ", Computer Physics Communications, 185 (2014), 3151-3155.
- 6
- J. Franco, and I. Gomez ``Symplectic explicit methods of Runge-Kutta-Nyström type for solving pertubed oscillators,Journal of Computational and Applied Mathematics, 260 (2014), 482-493.
- 7
- T. E. Simos, ``An exponentially-fitted Runge-Kutta method for the numerical integration of initial-value problems with periodic or oscillating solutions,Computer Physics Communications, 115 (1998), 1-8.
- 8
- N. Senu,``Runge-Kutta Nyström methods for solving oscillatory problems", Ph.D diss., Department of Mathematics, Faculty of Science, 43400 UPM Serdang, Malaysia, 2009.
- 9
- J. C. Butcher, ``Numerical methods for ordinary differential equations",Wiley & Sons LTD., England, 2008.
- 10
- Xinyuan Wu, Xiong You and Bin Wang, ``Structure preserving algorithms for oscillatory differential equations",Science Press Beijing & Springer-Verlag Berlin Heidelberg, Beijing, 2013.
- 11
- M. Mohamad,``Explicit Runge-Kutta Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation", M.sc diss., Department of Mathematics, Faculty of Science, 43400 UPM Serdang, Malaysia, 2013.
- 12
- Z. Anastassi and A. Kosti,``A 6(4) optimized embedded Runge-Kutta Nyström pair for the numerical solution of periodic problems ", Journal of Computational Applied Mathematics, 275 (2013), 311-320.
- 13
- N. Senu, M. Suleiman, F. Ismail and M. Othman, ``A singly diagonally implicit Runge-Kutta-Nyström method for solving oscillatory problems", IAENG International Journal of Applied Mathematics, 41 (2011), 155-161.
- 14
- A. Garcia, P. Martin, and A. B. Gonzalez et al, ``New methods for oscillatory problems based on classical codes", Applied Numerical Mathematics, 42 (2002), 141-157.
- 13
- N. Senu, M. Suleiman, and F. Ismail, ``An embedded explicit Runge-Kutta-Nyström method for solving oscillatory problems", Physica Scripta, 80 (2009), 015005.
- 15
- D. F. Papadopoulos, Z. Anastassi, A. Zacharias and T. E. Simos, ``A phase-fitted Runge-Kutta-Nyström method for the numerical solution of initial value problems with oscillating solutions", Computer Physics Communications, 180 (2009), 1839-1846.
How to Cite?
DOI: 10.12732/ijpam.v111i2.3 How to cite this paper?Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 111
Issue: 2
Pages: 165 - 178
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