Error Estimation by using Symmetrization and Efficient Implementation Scheme for 3-stage Gauss Method


  • Sara Syahrunnisaa Mustapha Mathematics Department, Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris, Tanjong Malim, Perak, Malaysia
  • Annie Gorgey Mathematics Department, Faculty of Science and Mathematics, Universiti Pendidikan Sultan Idris, Tanjong Malim, Perak, Malaysia
  • Gulshad Imran Professional and Continuing Education, Massey University, Auckland, New Zealand



variable stepsize setting, error estimation, local extrapolation, symmetrization


This research focuses on the implementation strategies by the implicit Runge-Kutta Gauss methods in solving Robertson problem using variable stepsize setting. This research considers ideas of implementation strategies by Hairer and Wanner (HW) and Gonzalez-Pinto, Montijano and Randez (GMR) schemes that uses a certain transformation matrix T to improve the efficiency of the numerical methods. Both implementations use simplified Newton iterations to solve the nonlinear algebraic equations for the implicit methods. These implementation strategies are compared with the modified Hairer and Wanner (MHW) scheme without using any transformation matrix T. The numerical methods considered are the implicit 3-stage Gauss (G3) method of order-6. The numerical results are given for Robertson problem which is a chemical reaction stiff problem. The variable stepsize setting is adapted in Matlab code that estimates the error using symmetrization technique.  Based on the numerical experiments, it is observed that GMR scheme is efficient by using the G3 method especially when the error estimation is obtained by using symmetrization technique instead of local extrapolation if compared with other schemes. In conclusion, GMR scheme is seen to be very robust in solving Robertson problem by the G3 method in terms of tolerance and computational time.


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Butcher, J. C. (2016). Numerical methods for Ordinary Differential Equations. United Kingdom: John Wiley & Sons.

Chan, R.P.K. & Gorgey, A. (2013). Active and passive symmetrization of Runge–Kutta Gauss methods. Applied Numerical Mathematics, 67, 64–77.

Cooper, G. J. & Butcher, J. C. (1983). An iteration scheme for implicit Runge-Kutta methods. IMA Journal of Numerical Analysis, 3(2), 127-140.

González-Pinto, S., González-Concepción, C. & Montijano, J. I. (1994). Iterative schemes for Gauss methods. Computers and Mathematics with Applications, 27(7), 67-81.

González-Pinto, S., Montijano, J. I. & Rández, L. (1995). Iterative schemes for three-stage implicit Runge-Kutta methods. Applied Numerical Mathematics, 17(4), 363–382.

González-Pinto, Domingo, H. A. & Montijano, J. I. (2020). Variable Step-Size Control Based on Two-Steps for Radau IIA Methods. ACM Transactions on Mathematical Software, 46(4), 1-24.

Gorgey, A. (2016). Extrapolation of symmetrized Runge-Kutta methods in the variable stepsize setting. International Journal of Applied Mathematics and Statistics, 55(2), 14-22.

Hairer, E. & Wanner, G. (1996). Solving ordinary differential equations II. London: Springer-Verlag Berlin Heidelberg.

Hairer, E. & Wanner, G. (1999). Stiff differential equations solved by Radau methods. Journal of Computational and Applied Mathematics, 111(1-2), 93-111.

Ismail, A. & Gorgey, A. (2013). Efficiency of Extrapolated Runge-Kutta methods in solving linear and nonlinear problems. Journal of Science and Mathematics Letters, 1, 1-8.

Ismail, A. & Gorgey, A. (2015). Behaviour of the Extrapolated Implicit Midpoint and Implicit Trapezoidal Rules With and Without Compensated Summation. Matematika, 31(1), 47–57.

Hindmarsh, A. C. (1980). LSODE and LSODI, two new initial value ordinary differential equation solvers. ACM Signum Newsletter, 15(4), 10-11.

Muhammad, M. H. & Gorgey, A. (2018). Investigation on the most efficient ways to solve the implicit equations for Gauss methods in the constant stepsize setting. Applied Mathematical Sciences, 12(2), 93-103.

Robertson, H. H. (1966). The solution of a set of reaction rate equations. Cambridge, Massachusetts: Academic Press. page 178-182.

Wang, P., Zhou, J., Wang, R. & Chen, J. (2017). New generalized variable stepsizes of the CQ algorithm for solving the split feasibility problem. Journal of Inequalities and Applications, 2017(1), 135.

Xu, P., Yuan, Z., Jian, W. & Zhao, W. (2015). Variable step-size method based on a reference separation system for source separation. Journal of Sensors, 2015. 964098.

Yang, X., Yang, Z. & Xiao, Y. (2020). Asymptotical mean-square stability of linear θ-methods for stochastic pantograph differential equations: variable stepsize and transformation approach. Unpublished manuscript. DOI: 10.22541/au.159023888.86381071




How to Cite

Mustapha, S. S., Gorgey, A., & Imran, G. (2021). Error Estimation by using Symmetrization and Efficient Implementation Scheme for 3-stage Gauss Method. Journal of Science and Mathematics Letters, 9, 36–44.