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

Abstract

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.