An Accurate Two-Step Optimized Hybrid Block Method for Integrating Stiff Differential Equations

An accurate two-step optimized hybrid block method is proposed for integrating stiff initial value problems of ordinary differential equations. The techniques of interpolation and collocation were applied to a power series polynomial for the derivation of the method using a three-parameter approximation of the hybrid points. The hybrid points were obtained by minimizing the local truncation error of the main method. The discrete schemes were produced as by-products of the continuous scheme and used to simultaneously solve initial value problems (IVPs) in block mode. The analysis of the basic properties of the method revealed that the schemes are self-starting, consistent, zero-stable, and A-stable. Furthermore, the analysis of the order of accuracy of the method showed that there is a gain of one order of accuracy in the main scheme where the optimization was carried out thereby enhancing the accuracy of the whole method. The accuracy of the method was ascertained using several numerical experiments. Comparison of the numerical results of the new method with those of the existing methods revealed that the newly developed method performed better than some of the existing hybrid block methods. Hence, the new method should be employed for the numerical solution of stiff ordinary differential equations to obtain more accurate results.