An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting

The research is aimed to find the most efficient implementation strategies by Gauss numericalmethods for solving stiff problems and the best error estimation in the variablestepsize setting. The numerical methods considered as a research methodology are the 2-stage(G2) and 3-stage (G3) implicit Rung...

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Main Author:	Sara Syahrunnisaa Mustapha
Format:	thesis
Language:	eng
Published:	2021
Subjects:	QA Mathematics
Online Access:	https://ir.upsi.edu.my/detailsg.php?det=5865
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spellingShingle	QA Mathematics Sara Syahrunnisaa Mustapha An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting
description	The research is aimed to find the most efficient implementation strategies by Gauss numericalmethods for solving stiff problems and the best error estimation in the variablestepsize setting. The numerical methods considered as a research methodology are the 2-stage(G2) and 3-stage (G3) implicit Runge-Kutta Gauss methods. Two strategies by Hairer andWanner (HW) and Gonzalez-Pinto, Montijano and Randez (GMR) schemes were implemented. Thevariable stepsize setting employed the simplified Newton is modified to fit according to HW andGMR schemes in solving the nonlinear algebraic systems of the equations. The errorestimation for the variablestepsize setting is computed using extrapolation technique with stepsizes h and h 2 .HW and GMR schemes used the transformation matrix T to improve the efficiency of the methods andalso compared with the modified Hairer and Wanner (MHW) schemewithout using any transformation matrix T . Findings showed that G2 method usingMHW scheme gave an efficient implementation in solving Kaps, Oreganator and HIRESproblems while for G3 method, it was efficient in solving Kaps, Brusselator, Oreganator, Van derPol and HIRES problems. In terms of error estimation, the G2 method gave the best error estimationfor Brusselator, Oreganator, Van der Pol and HIRES problems, while for the G3 method it wasefficient in solving Kaps, Brusselator, Oreganator, Van der Pol and HIRES problems, both byusing HW scheme. In conclusion, the MHW scheme without any transformation matrix T can be asefficient as the HW and GMR schemes by using the variable stepsize setting and the MHW scheme isrecommended in solving stiff problems. As for the implications, this research could be extendedto other different types of problems such as delay and fuzzyrential equations.
format	thesis
qualification_name
qualification_level	Master's degree
author	Sara Syahrunnisaa Mustapha
author_facet	Sara Syahrunnisaa Mustapha
author_sort	Sara Syahrunnisaa Mustapha
title	An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting
title_short	An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting
title_full	An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting
title_fullStr	An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting
title_full_unstemmed	An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting
title_sort	efficient implementation of runge-kutta gauss methods using variable stepsize setting
granting_institution	Universiti Pendidikan Sultan Idris
granting_department	Fakulti Sains dan Matematik
publishDate	2021
url	https://ir.upsi.edu.my/detailsg.php?det=5865
_version_	1747833233195663360
spelling	oai:ir.upsi.edu.my:58652021-04-16 An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting 2021 Sara Syahrunnisaa Mustapha QA Mathematics The research is aimed to find the most efficient implementation strategies by Gauss numericalmethods for solving stiff problems and the best error estimation in the variablestepsize setting. The numerical methods considered as a research methodology are the 2-stage(G2) and 3-stage (G3) implicit Runge-Kutta Gauss methods. Two strategies by Hairer andWanner (HW) and Gonzalez-Pinto, Montijano and Randez (GMR) schemes were implemented. Thevariable stepsize setting employed the simplified Newton is modified to fit according to HW andGMR schemes in solving the nonlinear algebraic systems of the equations. The errorestimation for the variablestepsize setting is computed using extrapolation technique with stepsizes h and h 2 .HW and GMR schemes used the transformation matrix T to improve the efficiency of the methods andalso compared with the modified Hairer and Wanner (MHW) schemewithout using any transformation matrix T . Findings showed that G2 method usingMHW scheme gave an efficient implementation in solving Kaps, Oreganator and HIRESproblems while for G3 method, it was efficient in solving Kaps, Brusselator, Oreganator, Van derPol and HIRES problems. In terms of error estimation, the G2 method gave the best error estimationfor Brusselator, Oreganator, Van der Pol and HIRES problems, while for the G3 method it wasefficient in solving Kaps, Brusselator, Oreganator, Van der Pol and HIRES problems, both byusing HW scheme. In conclusion, the MHW scheme without any transformation matrix T can be asefficient as the HW and GMR schemes by using the variable stepsize setting and the MHW scheme isrecommended in solving stiff problems. As for the implications, this research could be extendedto other different types of problems such as delay and fuzzyrential equations. 2021 thesis https://ir.upsi.edu.my/detailsg.php?det=5865 https://ir.upsi.edu.my/detailsg.php?det=5865 text eng closedAccess Masters Universiti Pendidikan Sultan Idris Fakulti Sains dan Matematik Ababneh, O. Y. & Ahmad, R. (2009). Construction of third-order diagonal implicitRunge-Kutta methods for stiff problems. Chinese Physics Letters, 26(8), 080503.Abia, L. & Sanz-Serna, J. M. (1993). Partitioned Runge-Kutta methods for separable Hamiltonianproblems. Mathematics of Computation, 60(202), 617-634.Agam, S. A. & Yahaya, Y. A. (2014). A highly efficient implicit Runge-Kutta method for first orderordinary differential equations. 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An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting

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