An Eighth-Stage Implicit Runge-Kutta Type Scheme for First-Order Ordinary Differential Equations

Abstract

We develop an eighth-stage implicit Runge-Kutta (EIRK) type scheme for solving first-order initial-value problems (IVPs) in ordinary differential equations (ODEs). Through the multistep collocation approach, we obtain a one-step continuous hybrid solution, which is evaluated at certain points of interest to generate seven discrete schemes that are used to formulate a block hybrid method. The resultant block method is then reformulated into the new EIRK scheme. Analysis of the basic properties of the novel EIRK indicates that the method is of order eight, hence consistent, indicating an improvement over methods of lower orders, leading to better accuracy. Furthermore, linear stability analysis reveals that this method is A-stable, as such is a viable candidate for stiff IVPs. The numerical examples considered using the EIRK show that this proposed method yielded smaller absolute errors, implying better accuracy, as the tables reveal when compared with similar existing methods in the literature, hence should be preferred for such a class of problems