Abstract.

This paper presents a novel class of explicit five-step block methods tailored for the efficient and accurate solution of second-order initial value problems (IVPs), particularly those characterized by oscillatory behavior. The proposed methods incorporate non-symmetric multistep structures and follows elimination of phase lag and amplification error, along with their first derivatives. A rigorous stability analysis confirms the robustness of the schemes across a wide range of frequencies. Numerical experiments including ordinary as well as partial differential equations, further substantiate the effectiveness of the approach, demonstrating its potential as a powerful tool for solving second-order differential systems.

Keywords:

numerical solution, initial value problems (IVPs), Block methods, phase-fitting.