Variable stepsize construction of a two-step optimized hybrid block method with relative stability

Abstract

Several numerical techniques for solving initial value problems arise in physical and natural sciences. In many cases, these problems require numerical treatment to achieve the required solution. However, in today’s modern era, numerical algorithms must be cost-effective with suitable convergence and stability features. At least the fifth-order convergent two-step optimized hybrid block method recently proposed in the literature is formulated in this research work with its variable stepsize approach for numerically solving first- and higher-order initial-value problems in ordinary differential equations. It has been constructed using a continuous approximation achieved through interpolation and collocation techniques at two intra-step points chosen by optimizing the local truncation errors of the main formulae. The theoretical analysis, including order stars for the relative stability, is considered. Both fixed and variable stepsize approaches are presented to observe the superiority of the latter approach. When tested on challenging differential systems, the method gives better accuracy, as revealed by the efficiency plots and the error distribution tables, including the machine time measured in seconds.