Implementation of a High-Order Diagonally Implicit Block Backward Differentiation Method Incorporating Off-Step Points for Solving Stiff ODEs

Abstract

This work focuses on the derivation and implementation of a 2-point diagonally implicit block backward differentiation formula that integrates off-step points to solve first-order stiff initial value problems (IVPs). The primary architecture of the proposed block method involves the simultaneous computation of two approximate solution values, ​ and ​, alongside two strategically positioned off-step points,  and ​, at each iteration stage. A rigorous evaluation of the scheme's structural properties confirms that it achieves a true seventh-order algebraic convergence. Furthermore, stability analysis demonstrates that the method is both zero-stable and absolute-stable, making it exceptionally well-suited for the demanding demands of stiff dynamical systems. To evaluate its practical performance, the scheme was deployed against several classic first-order stiff IVPs, and the outcomes were benchmarked against existing state-of-the-art numerical solvers. The empirical results demonstrate that the proposed method consistently outperforms competing schemes, offering a distinct advantage in minimizing scale error while significantly reducing computational execution time. Consequently, this new framework stands as a highly accurate and computationally efficient alternative for integrating first-order stiff ordinary differential equations.