Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems
In this paper, order conditions for Runge-Kutta-Nyström (RKN) method for solving second order linear ordinary differential equations (LODEs) are derived up to order six. Based on the order conditions, a new fifth order four-stage RKN method which is specially designed for the integration of second o...
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| Format: | Article |
| Language: | English |
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Pushpa Publishing House
2016
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| Online Access: | http://psasir.upm.edu.my/id/eprint/54314/1/Fifth%20order%20Runge-Kutta-Nystr%C3%B6m%20methods%20.pdf |
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| author | Salih, Mohammed M. Ismail, Fudziah Senu, Norazak |
| author_facet | Salih, Mohammed M. Ismail, Fudziah Senu, Norazak |
| author_sort | Salih, Mohammed M. |
| collection | UPM |
| description | In this paper, order conditions for Runge-Kutta-Nyström (RKN) method for solving second order linear ordinary differential equations (LODEs) are derived up to order six. Based on the order conditions, a new fifth order four-stage RKN method which is specially designed for the integration of second order linear ordinary differential equations (LODEs) is constructed with first same as last (FSAL) property. Then we phase-fitted the method such that it has zero phase-lag and zero dissipation. Phase-lag or dispersion error is the angle between the true and the approximated solutions, whereas dissipation is the distance of the computed solution from the standard cyclic solution. A set of test problems are used to validate the method and numerical results show that the phase-fitted method produced smaller global error compared to the original method as well as other existing RKN method in the scientific literature. |
| first_indexed | 2024-03-06T09:20:18Z |
| format | Article |
| id | upm.eprints-54314 |
| institution | Universiti Putra Malaysia |
| language | English |
| last_indexed | 2024-03-06T09:20:18Z |
| publishDate | 2016 |
| publisher | Pushpa Publishing House |
| record_format | dspace |
| spelling | upm.eprints-543142018-03-13T04:15:28Z http://psasir.upm.edu.my/id/eprint/54314/ Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems Salih, Mohammed M. Ismail, Fudziah Senu, Norazak In this paper, order conditions for Runge-Kutta-Nyström (RKN) method for solving second order linear ordinary differential equations (LODEs) are derived up to order six. Based on the order conditions, a new fifth order four-stage RKN method which is specially designed for the integration of second order linear ordinary differential equations (LODEs) is constructed with first same as last (FSAL) property. Then we phase-fitted the method such that it has zero phase-lag and zero dissipation. Phase-lag or dispersion error is the angle between the true and the approximated solutions, whereas dissipation is the distance of the computed solution from the standard cyclic solution. A set of test problems are used to validate the method and numerical results show that the phase-fitted method produced smaller global error compared to the original method as well as other existing RKN method in the scientific literature. Pushpa Publishing House 2016-08 Article PeerReviewed text en http://psasir.upm.edu.my/id/eprint/54314/1/Fifth%20order%20Runge-Kutta-Nystr%C3%B6m%20methods%20.pdf Salih, Mohammed M. and Ismail, Fudziah and Senu, Norazak (2016) Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems. Far East Journal of Applied Mathematics, 95 (2). pp. 141-156. ISSN 0972-0960 http://www.pphmj.com/abstract/10193.htm Runge-Kutta-Nyström; Phase-fitted; Oscillatory; Linear ODEs 10.17654/AM095020141 |
| spellingShingle | Runge-Kutta-Nyström; Phase-fitted; Oscillatory; Linear ODEs Salih, Mohammed M. Ismail, Fudziah Senu, Norazak Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems |
| title | Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems |
| title_full | Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems |
| title_fullStr | Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems |
| title_full_unstemmed | Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems |
| title_short | Fifth order Runge-Kutta-Nyström methods for solving linear second order oscillatory problems |
| title_sort | fifth order runge kutta nystrom methods for solving linear second order oscillatory problems |
| topic | Runge-Kutta-Nyström; Phase-fitted; Oscillatory; Linear ODEs |
| url | http://psasir.upm.edu.my/id/eprint/54314/1/Fifth%20order%20Runge-Kutta-Nystr%C3%B6m%20methods%20.pdf |
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