Implicit 7-stage Tenth Order Runge-Kutta Methods Based on Gauss-Kronrod-Lobatto Quadrature Formula

In this paper, four new implicit Runge-Kutta methods which based on 7- point Gauss-Kronrod-Lobatto quadrature formula were developed. The resulting im- plicit 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 or- der 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 anal- yses 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 hav- ing 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. Last but not least, all the proposed methods were implemented to solve two real-world problems i.e. the Van der Pol oscillator and the Brusselator. The numerical solutions which generated by the proposed methods were found to be comparable to the numerical solutions found in the existing literature.