Some eight - step implicit linear multistep methods of order ten

Abstract

In this paper we analyze the Taylor series method of deriving linear multistep methods through expansion of the linear difference operator ℒ[𝑦(𝑥);ℎ]=Σ[𝛼𝑗𝑦(𝑥+𝑗ℎ)−ℎ𝛽𝑗𝑦′(𝑥+𝑗ℎ)] where 𝑦(𝑥) is an arbitrary function continuously differentiable on [𝑎,𝑏]. The resulting constant expressions 𝐷𝑖 (𝑖=0(1)11) are expanded and solved accordingly. By a careful and judicious assignment of appropriate values to the free parameters, we obtain two eight – step implicit linear multistep schemes of optimal order (in this case order ten). The schemes are shown to be consistent and zero – stable; thereby establishing their convergence. In order to affirm their efficacy and reliability, the schemes are applied to sample initial value problems and the results compared to exact solutions. The negligibility of the exhibited errors further confirmed their usefulness.