An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems

This research is aimed to produce an efficient implementation technique for the 2-stage(G2) and 3-stage (G3) implicit Runge-Kutta Gauss methods in solving mathematical stiff problems.Both methods are constructed by using Maple software and have been implemented by using Matlabsoftware numerically. T...

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Main Author:	Mohd Hafizul Muhammad
Format:	thesis
Language:	eng
Published:	2018
Subjects:	QA Mathematics
Online Access:	https://ir.upsi.edu.my/detailsg.php?det=5755
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institution	Universiti Pendidikan Sultan Idris
collection	UPSI Digital Repository
language	eng
topic	QA Mathematics
spellingShingle	QA Mathematics Mohd Hafizul Muhammad An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems
description	This research is aimed to produce an efficient implementation technique for the 2-stage(G2) and 3-stage (G3) implicit Runge-Kutta Gauss methods in solving mathematical stiff problems.Both methods are constructed by using Maple software and have been implemented by using Matlabsoftware numerically. This research applied four imple- mentation strategies which are full Newtonwithout compensated summation (FNWSN), full Newton with compensated summation (FNCS), simplifiedNewton without com- pensated summation (SNWCS) and simplified Newton with compensated summation(SNCS). Comparison have been done with the implementations of Hairer and Wanner scheme, Cooper andButcher scheme, and Gonzlez scheme. Results for stiff test prob- lems showed that SNCS is the mostefficient technique in solving some real life mathe- matical problems such as the Kepler,Oregonator, Van der Pol, HIRES and Brusselator problems. According to the numerical results, theimplementation of G2 using SNCS by the Hairer and Wanner scheme is the most efficient technique forsolving Kepler and Brusselator problems, while SNCS by the Gonzlez scheme is the most efficienttechnique for solving other problems. On the contrary for G3, SNCS by the Hairer and Wanner schemegives the most efficient technique for solving Kepler and Van der Pol problems, while SNCS by theGonzlez scheme gives the most efficient technique for solving other problems. In conclusion, forboth G2 and G3 methods, SNCS plays an important role to improve the efficiency of implicitRunge-Kutta Gauss methods in solv- ing mathematical stiff problems. As for the implications, theimplementation technique used in this research can be extended during tertiary education on thesubject numer- ical ordinary differential equations that focusses on implementation schemes byotherresearchers as well as to some other implicit Runge-Kutta methods.
format	thesis
qualification_name
qualification_level	Master's degree
author	Mohd Hafizul Muhammad
author_facet	Mohd Hafizul Muhammad
author_sort	Mohd Hafizul Muhammad
title	An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems
title_short	An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems
title_full	An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems
title_fullStr	An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems
title_full_unstemmed	An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems
title_sort	efficient implementation technique for implicit runge-kutta gauss methods in solving mathematical stiff problems
granting_institution	Universiti Pendidikan Sultan Idris
granting_department	Fakulti Sains dan Matematik
publishDate	2018
url	https://ir.upsi.edu.my/detailsg.php?det=5755
_version_	1747833223786790912
spelling	oai:ir.upsi.edu.my:57552021-04-05 An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems 2018 Mohd Hafizul Muhammad QA Mathematics This research is aimed to produce an efficient implementation technique for the 2-stage(G2) and 3-stage (G3) implicit Runge-Kutta Gauss methods in solving mathematical stiff problems.Both methods are constructed by using Maple software and have been implemented by using Matlabsoftware numerically. This research applied four imple- mentation strategies which are full Newtonwithout compensated summation (FNWSN), full Newton with compensated summation (FNCS), simplifiedNewton without com- pensated summation (SNWCS) and simplified Newton with compensated summation(SNCS). Comparison have been done with the implementations of Hairer and Wanner scheme, Cooper andButcher scheme, and Gonzlez scheme. Results for stiff test prob- lems showed that SNCS is the mostefficient technique in solving some real life mathe- matical problems such as the Kepler,Oregonator, Van der Pol, HIRES and Brusselator problems. According to the numerical results, theimplementation of G2 using SNCS by the Hairer and Wanner scheme is the most efficient technique forsolving Kepler and Brusselator problems, while SNCS by the Gonzlez scheme is the most efficienttechnique for solving other problems. On the contrary for G3, SNCS by the Hairer and Wanner schemegives the most efficient technique for solving Kepler and Van der Pol problems, while SNCS by theGonzlez scheme gives the most efficient technique for solving other problems. In conclusion, forboth G2 and G3 methods, SNCS plays an important role to improve the efficiency of implicitRunge-Kutta Gauss methods in solv- ing mathematical stiff problems. As for the implications, theimplementation technique used in this research can be extended during tertiary education on thesubject numer- ical ordinary differential equations that focusses on implementation schemes byotherresearchers as well as to some other implicit Runge-Kutta methods. 2018 thesis https://ir.upsi.edu.my/detailsg.php?det=5755 https://ir.upsi.edu.my/detailsg.php?det=5755 text eng closedAccess Masters Universiti Pendidikan Sultan Idris Fakulti Sains dan Matematik Agam, S. A. and Yahaya, Y. A. (2014). A highly efficient implicit Runge-Kutta methodfor first order ordinary differential equations. African Journal of Mathematics and ComputerScience Research, 7(5): 5560.Bashforth, F. (1883). 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An efficient implementation technique for implicit Runge-Kutta Gauss methods in solving mathematical stiff problems

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