Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations

This thesis deals with the derivation of diagonally implicit Runge-Kutta (DIRK) methods which are specially designed for the integration of linear ordinary differential equations (LODEs). The restriction to LODEs with constant coefficients reduces the number of order equations which the coefficients...

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Main Author:	Che Jawias, Nur Izzati
Format:	Thesis
Language:	English English
Published:	2009
Online Access:	http://psasir.upm.edu.my/id/eprint/5729/ http://psasir.upm.edu.my/id/eprint/5729/1/A__FS_2009_18.pdf

id	upm-5729
recordtype	eprints
spelling	upm-57292013-05-27T07:24:44Z http://psasir.upm.edu.my/id/eprint/5729/ Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations Che Jawias, Nur Izzati This thesis deals with the derivation of diagonally implicit Runge-Kutta (DIRK) methods which are specially designed for the integration of linear ordinary differential equations (LODEs). The restriction to LODEs with constant coefficients reduces the number of order equations which the coefficients of Runge-Kutta (RK) methods must satisfy. This freedom is used to construct new methods which are more efficient compared to the conventional RK methods. Having achieved a particular order of accuracy, the best strategy for practical purposes would be to choose the coefficients of the RK methods such that the error norm is minimized. The free parameters chosen are obtained from the minimized error norm. This resulted in methods which are almost one order higher than the actual order. In this thesis we construct a fourth order DIRK method without taking into account the error norm. We also construct fourth and fifth order DIRK methods using the minimized error norm. The stability aspects of the methods are investigated by finding the stability polynomials of the methods, which are then solved to obtain the stability regions using MATHEMATICA package. The methods are found to have bigger regions of stability compared to the explicit Runge-Kutta (ERK) methods of the same type (designed for the integration of LODEs). Later, we built codes using C++ programming based on the methods. Sets of test problems on linear ordinary differential equations are used to validate the methods and numerical results show that the new methods produce smaller global error compared to ERK methods. From the stability regions and numerical results obtained, we can conclude that the new DIRK methods are more stable and more accurate compared to the explicit one. Higher order methods also gives better result compared to lower order methods. 2009 Thesis NonPeerReviewed application/pdf en http://psasir.upm.edu.my/id/eprint/5729/1/A__FS_2009_18.pdf Che Jawias, Nur Izzati (2009) Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations. Masters thesis, Universiti Putra Malaysia. English
repository_type	Digital Repository
institution_category	Local University
institution	Universiti Putra Malaysia
building	UPM Institutional Repository
collection	Online Access
language	English English
description	This thesis deals with the derivation of diagonally implicit Runge-Kutta (DIRK) methods which are specially designed for the integration of linear ordinary differential equations (LODEs). The restriction to LODEs with constant coefficients reduces the number of order equations which the coefficients of Runge-Kutta (RK) methods must satisfy. This freedom is used to construct new methods which are more efficient compared to the conventional RK methods. Having achieved a particular order of accuracy, the best strategy for practical purposes would be to choose the coefficients of the RK methods such that the error norm is minimized. The free parameters chosen are obtained from the minimized error norm. This resulted in methods which are almost one order higher than the actual order. In this thesis we construct a fourth order DIRK method without taking into account the error norm. We also construct fourth and fifth order DIRK methods using the minimized error norm. The stability aspects of the methods are investigated by finding the stability polynomials of the methods, which are then solved to obtain the stability regions using MATHEMATICA package. The methods are found to have bigger regions of stability compared to the explicit Runge-Kutta (ERK) methods of the same type (designed for the integration of LODEs). Later, we built codes using C++ programming based on the methods. Sets of test problems on linear ordinary differential equations are used to validate the methods and numerical results show that the new methods produce smaller global error compared to ERK methods. From the stability regions and numerical results obtained, we can conclude that the new DIRK methods are more stable and more accurate compared to the explicit one. Higher order methods also gives better result compared to lower order methods.
format	Thesis
author	Che Jawias, Nur Izzati
spellingShingle	Che Jawias, Nur Izzati Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations
author_facet	Che Jawias, Nur Izzati
author_sort	Che Jawias, Nur Izzati
title	Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations
title_short	Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations
title_full	Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations
title_fullStr	Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations
title_full_unstemmed	Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations
title_sort	diagonally implicit runge-kutta methods for solving linear ordinary differential equations
publishDate	2009
url	http://psasir.upm.edu.my/id/eprint/5729/ http://psasir.upm.edu.my/id/eprint/5729/1/A__FS_2009_18.pdf
first_indexed	2018-09-07T12:22:12Z
last_indexed	2018-09-07T12:22:12Z
_version_	1610951355777679360

Diagonally Implicit Runge-Kutta Methods For Solving Linear Ordinary Differential Equations

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