An efficient implementation of Runge-Kutta Gauss methods using variable stepsize setting - Sara Syahrunnisaa Mustapha (2021)
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.