Abstract

In this paper, the optimal 8--step linear multistep method for solving $y^{\prime}=f(x,y)$ is constructed and implemented. The construction was carried out using the technique based on the Taylor expansion of $y(x + jh)$ and $y^{\prime}(x + jh)$ about $x + t h$, where \emph{t} need not necessarily be an integer. The consistency, stability and convergence of the proposed method are investigated. To investigate the accuracy of the method, a comparison with the classical 8-stage Runge--Kutta method is carried out on two numerical examples. The results obtained by the constructed method are accurate up to certain degrees and compete favourably with those produced by the classical 8-stage Runge--Kutta method.

Keywords: Linear Multistep Method, Optimal, Consistency, Stability, and Accuracy