Optimized two-step second derivative methods for the solutions of stiff systems

Abstract

In this research article, a pair of optimized two-step second derivative methods is derived and implemented on stiff systems. The influence of equidistant and non-equidistant hybrid points spacing on the performance of the methods derived is investigated. Firstly, the methods are derived using interpolation and collocation of a finite power series at some selected grid points. This leads to the formation of a system of nonlinear equations, which are then solved for the unknown parameters to obtain a continuous second derivative scheme. The by-products of the evaluation of the continuous second derivative scheme lead to the development of discrete methods in the form of blocks. Secondly, the basic properties of the methods derived were analysed. Numerical results were generated to investigate the influence of equidistant and non-equidistant hybrid points spacing on the performance of the methods on stiff systems. The results so obtained clearly showed that the two-step second derivative method with equidistant hybrid point spacing performed better than the two-step second derivative method with non-equidistant hybrid point spacing. This implies that equidistant hybrid point spacing enhances the accuracy of a method as opposed to non-equidistant hybrid point spacing.

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