Stable and Efficient Second Derivative Block Hybrid Method for the Solution of Stiff Systems of Ordinary Differential Equations

Abstract

 Stiff systems of ordinary differential equations (ODEs) often appear in scientific and engineering applications where solutions exhibit rapid variation, making their study highly significant. Traditional numerical approaches, including explicit Runge–Kutta methods and several multistep schemes, usually face stability challenges when applied to such problems. To maintain stability, very small step sizes are required, which greatly increases computational cost. To address this limitation, this work proposes a new second-derivative block hybrid method designed for the efficient solution of stiff ODEs. The method relies on a power series expansion that is both interpolated and collocated at selected points to generate a continuous block scheme. By incorporating off-grid points together with second derivatives, the technique achieves high-order accuracy while ensuring zero-stability, consistency, and convergence. The developed scheme is characterized by a broad region of absolute stability and maintains a uniform order of twelve. Numerical experiments carried out on well-known stiff test problems highlight the reliability and efficiency of the method, producing errors far smaller than those obtained with conventional techniques. The results provide clear evidence of the method’s ability to handle the computational difficulties associated with stiff systems and establish it as a strong alternative to existing approaches. This contribution enhances the progress of more advanced numerical strategies capable of performing effectively in real-world fields such as chemical reaction dynamics, systems biology, and control processes.