Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations

Block diagonally implicit Runge-Kutta (BDIRK) method of second order is derived using Butcher analysis and equi-distribution of errors (EDE) approach to outperform standard Runge-Kutta formulae. BDIRK method produces simultaneously approximation to the solution of the initial value problem (IVP) at...

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Main Author:	Rahim, Yong Faezah
Format:	Thesis
Language:	English English
Published:	2004
Subjects:	Differential equations - Numerical solutions Runge-Kutta formulas
Online Access:	http://psasir.upm.edu.my/id/eprint/483/1/549720_FSAS_2004_14.pdf
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spelling	my-upm-ir.4832013-05-27T06:48:39Z Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations 2004-03 Rahim, Yong Faezah Block diagonally implicit Runge-Kutta (BDIRK) method of second order is derived using Butcher analysis and equi-distribution of errors (EDE) approach to outperform standard Runge-Kutta formulae. BDIRK method produces simultaneously approximation to the solution of the initial value problem (IVP) at a block of two points and . Numerical results are given to illustrate the performance of the method and comparison is made to the standard Runge-Kutta method. In this thesis, we also derived block method using embedded diagonally implicit Runge-Kutta method. The conventional one step embedded diagonally implicit Runge-Kutta (DIRK) 3(2) method which is known to be reliable and has been tested extensively is used to approximate the solution at the first point of the block, and also to generate the solution at the second point of the block. The efficiency of the method is supported by some numerical results. Finally, block embedded diagonally implicit Runge-kutta method is extended from approximating solutions at two points to three points simultaneously by increasing the order of the method. The results indicate that the block methods are competitive, both in terms of efficiency and reliability with conventional Runge-Kutta method. The stability polynomial for all the methods are obtained and their regions of stability are presented. Based on the numerical results and the stability regions it can be said that the method is suitable for solving stiff differential equations. Differential equations - Numerical solutions Runge-Kutta formulas 2004-03 Thesis http://psasir.upm.edu.my/id/eprint/483/ http://psasir.upm.edu.my/id/eprint/483/1/549720_FSAS_2004_14.pdf application/pdf en public masters Universiti Putra Malaysia Differential equations - Numerical solutions Runge-Kutta formulas Faculty of Environmental Studies English
institution	Universiti Putra Malaysia
collection	PSAS Institutional Repository
language	English English
topic	Differential equations - Numerical solutions Runge-Kutta formulas
spellingShingle	Differential equations - Numerical solutions Runge-Kutta formulas Rahim, Yong Faezah Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations
description	Block diagonally implicit Runge-Kutta (BDIRK) method of second order is derived using Butcher analysis and equi-distribution of errors (EDE) approach to outperform standard Runge-Kutta formulae. BDIRK method produces simultaneously approximation to the solution of the initial value problem (IVP) at a block of two points and . Numerical results are given to illustrate the performance of the method and comparison is made to the standard Runge-Kutta method. In this thesis, we also derived block method using embedded diagonally implicit Runge-Kutta method. The conventional one step embedded diagonally implicit Runge-Kutta (DIRK) 3(2) method which is known to be reliable and has been tested extensively is used to approximate the solution at the first point of the block, and also to generate the solution at the second point of the block. The efficiency of the method is supported by some numerical results. Finally, block embedded diagonally implicit Runge-kutta method is extended from approximating solutions at two points to three points simultaneously by increasing the order of the method. The results indicate that the block methods are competitive, both in terms of efficiency and reliability with conventional Runge-Kutta method. The stability polynomial for all the methods are obtained and their regions of stability are presented. Based on the numerical results and the stability regions it can be said that the method is suitable for solving stiff differential equations.
format	Thesis
qualification_level	Master's degree
author	Rahim, Yong Faezah
author_facet	Rahim, Yong Faezah
author_sort	Rahim, Yong Faezah
title	Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations
title_short	Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations
title_full	Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations
title_fullStr	Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations
title_full_unstemmed	Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations
title_sort	block diagonally implicit runge-kutta method for solving ordinary differential equations
granting_institution	Universiti Putra Malaysia
granting_department	Faculty of Environmental Studies
publishDate	2004
url	http://psasir.upm.edu.my/id/eprint/483/1/549720_FSAS_2004_14.pdf
_version_	1747810231302225920

Block Diagonally Implicit Runge-Kutta Method for Solving Ordinary Differential Equations

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