An Accuracy-preserving Block Hybrid Algorithm for the Integration of Second-order Physical Systems with Oscillatory Solutions

Abstract

It is a known fact that in most cases, to integrate an oscillatory problem, higher order A-stable methods are often needed. This is because such problems are characterized by stiffness, chaos and damping, thus making them tedious to solve. However, in this research, an accuracy-preserving relatively lower order Block Hybrid Algorithm (BHA) is proposed for solution of second-order physical systems with oscillatory solutions. The sixth order algorithm was derived using interpolation and collocation of power series within a single step interval [tn; tn+1]. In order to circumvent the Dahlquist-barrier and also obtain an accuracy-preserving algorithm, four o-step points were incorporated within the single step interval. A number of special cases of oscillatory problems were solved using the proposed method and the results obtained clearly showed that it outperformed other existing methods we compared our results with even though the BHA is of lower order relative to such methods. Some of the second-order physical systems considered were the Kepler, Bessel and damped problems. Some important properties of the BHA were also analyzed and the results of the analysis showed that it is consistent, zero-stable and convergent