Implicit Runge-Kutta Methods Based on Gauss-Kronrod-Lobatto Quadrature Formulae


  • Teh Yuan Ying Universiti Utara Malaysia
  • Nazeeruddin Yaacob Universiti Teknologi Malaysia



In this paper, four new implicit Runge-Kutta methods which based on 7-point Gauss-Kronrod-Lobatto quadrature formula were developed. The resulting implicit methods were 7-stage tenth order Gauss-Kronrod-Lobatto III (GKLM(7,10)-III), 7-stage tenth order Gauss-Kronrod-Lobatto IIIA (GKLM(7,10)-IIIA), 7-stage tenth order Gauss-Kronrod-Lobatto IIIB (GKLM(7,10)-IIIB) and 7-stage tenth order Gauss-Kronrod-Lobatto IIIC (GKLM(7,10)-IIIC). Each of these methods required 7 function of evaluations at each integration step and gave accuracy of order 10. Theoretical analyses showed that the stage order for GKLM(7,10)-III, GKLM(7,10)-IIIA, GKLM(7,10)-IIIB and GKLM(7,10)-IIIC are 6, 7, 3 and 4, respectively. GKLM(7,10)-IIIC possessed the strongest stability condition i.e. L-stability, followed by GKLM(7,10)-IIIA and GKLM(7,10)-IIIB which both possessed A-stability, and lastly GKLM(7,10)-III having finite region of absolute stability. Numerical experiments compared the accuracy of these four implicit methods and the classical 5-stage tenth order Gauss-Legendre method in solving some test problems. Numerical results revealed that, GKLM(7,10)-IIIA was the most accurate method in solving a scalar stiff problem. All the proposed methods were found to have comparable accuracy and more accurate than the 5-stage tenth order Gauss-Legendre method in solving a two-dimensional stiff problem.

Author Biographies

Teh Yuan Ying, Universiti Utara Malaysia

School of Quantitative Sciences, UUM College of Arts and Sciences, Universiti Utara Malaysia, 06010 UUM Sintok, Kedah Darul Aman.

Nazeeruddin Yaacob, Universiti Teknologi Malaysia

Department of Mathematics, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor Darul Ta'zim.




How to Cite

Yuan Ying, T., & Yaacob, N. (2015). Implicit Runge-Kutta Methods Based on Gauss-Kronrod-Lobatto Quadrature Formulae. MATEMATIKA, 31(1), 93–109.