Convergence analysis of a hybrid super class of block backward differentiation formula for integrating stiff IVP

Abstract

This research introduces a novel, seventh-order numerical scheme specifically engineered for the integration of first-order stiff initial value problems (IVPs). To establish the mathematical validity and reliability of the proposed method, a rigorous analysis of its structural and asymptotic properties was conducted. The fundamental necessary and sufficient conditions governing the convergence of linear multistep methods were systematically evaluated. Specifically, the order of accuracy and the error constant were analytically derived, confirming a true seventh-order algebraic convergence. Furthermore, an investigation into the stability architecture demonstrates that the scheme is both consistent and zero-stable. According to the Dahlquist equivalence theorem, the simultaneous satisfaction of these two properties guarantees the convergence of the method. Given its robust stability region and high-order precision, this scheme offers a computationally efficient and highly accurate alternative to existing solvers for handling stiff dynamical systems.