Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation

An explicit Runge-Kutta-Nyström (RKN) method with high order dispersion (phase-lag) and dissipation (amplification error) properties is studied for the integration of initial-value problems (IVP) of second-order ordinary differential equations (ODEs) possessing oscillating solutions. The constructio...

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Main Author:	Mohamad, Munirah
Format:	Thesis
Language:	English
Published:	2013
Subjects:	Numerical analysis Runge-Kutta formulas
Online Access:	http://psasir.upm.edu.my/id/eprint/67461/1/FS%202013%2070%20IR.pdf
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spelling	my-upm-ir.674612019-03-14T02:42:00Z Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation 2013-07 Mohamad, Munirah An explicit Runge-Kutta-Nyström (RKN) method with high order dispersion (phase-lag) and dissipation (amplification error) properties is studied for the integration of initial-value problems (IVP) of second-order ordinary differential equations (ODEs) possessing oscillating solutions. The constructions of RKN methods for constant step size and embedded RKN pair for variable step size have been derived. The effects of dispersion and dissipation relations are tested on homogeneous and non-homogenous test problems which have oscillatory solutions.Derivation of symplectic explicit Runge-Kutta-Nyström method is studied for Hamiltonian system with oscillating solutions. Symplectic methods can be more efficient than non-symplectic methods for long interval of integration. Numerical results show that the symplectic methods with high order of dispersion are more efficient for solving second order ordinary differential equations. The new fourth and fifth order explicit RKN methods with dispersion (phase-fitted) and dissipation (amplification-fitted) of order infinity have been derived. The fifth order explicit RKN methods is divided into two parts; methods with phase-fitted and methods with both phase-fitted and amplification-fitted. In this thesis, the phase-fitted methods are derived based on symplectic method by Sharp and method derived in this thesis. For fifth order phase-fitted and amplification-fitted consists of four- and five-stage RKN methods with First Same As Last (FSAL) technique. Numerical results show that our methods are much more efficient than existing method with the same algebraic order. In conclusion, we have derived explicit RKN methods with dispersion and dissipation for solving second-order ODEs that possessing oscillatory solutions. The dispersion constant, dissipation constant and local truncation error (LTE) terms are also calculated. The homogeneous and non-homogenous test problems with oscillatory solutions are used to prove the efficiency of our methods. Numerical analysis Runge-Kutta formulas 2013-07 Thesis http://psasir.upm.edu.my/id/eprint/67461/ http://psasir.upm.edu.my/id/eprint/67461/1/FS%202013%2070%20IR.pdf text en public masters Universiti Putra Malaysia Numerical analysis Runge-Kutta formulas
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English
topic	Numerical analysis Runge-Kutta formulas
spellingShingle	Numerical analysis Runge-Kutta formulas Mohamad, Munirah Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation
description	An explicit Runge-Kutta-Nyström (RKN) method with high order dispersion (phase-lag) and dissipation (amplification error) properties is studied for the integration of initial-value problems (IVP) of second-order ordinary differential equations (ODEs) possessing oscillating solutions. The constructions of RKN methods for constant step size and embedded RKN pair for variable step size have been derived. The effects of dispersion and dissipation relations are tested on homogeneous and non-homogenous test problems which have oscillatory solutions.Derivation of symplectic explicit Runge-Kutta-Nyström method is studied for Hamiltonian system with oscillating solutions. Symplectic methods can be more efficient than non-symplectic methods for long interval of integration. Numerical results show that the symplectic methods with high order of dispersion are more efficient for solving second order ordinary differential equations. The new fourth and fifth order explicit RKN methods with dispersion (phase-fitted) and dissipation (amplification-fitted) of order infinity have been derived. The fifth order explicit RKN methods is divided into two parts; methods with phase-fitted and methods with both phase-fitted and amplification-fitted. In this thesis, the phase-fitted methods are derived based on symplectic method by Sharp and method derived in this thesis. For fifth order phase-fitted and amplification-fitted consists of four- and five-stage RKN methods with First Same As Last (FSAL) technique. Numerical results show that our methods are much more efficient than existing method with the same algebraic order. In conclusion, we have derived explicit RKN methods with dispersion and dissipation for solving second-order ODEs that possessing oscillatory solutions. The dispersion constant, dissipation constant and local truncation error (LTE) terms are also calculated. The homogeneous and non-homogenous test problems with oscillatory solutions are used to prove the efficiency of our methods.
format	Thesis
qualification_level	Master's degree
author	Mohamad, Munirah
author_facet	Mohamad, Munirah
author_sort	Mohamad, Munirah
title	Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation
title_short	Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation
title_full	Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation
title_fullStr	Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation
title_full_unstemmed	Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation
title_sort	explicit runge-kutta-nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation
granting_institution	Universiti Putra Malaysia
publishDate	2013
url	http://psasir.upm.edu.my/id/eprint/67461/1/FS%202013%2070%20IR.pdf
_version_	1747812476694560768

Explicit Runge-Kutta-Nyström methods with high order dispersion and dissipation for solving oscillatory second order ordinary differential equation

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