A Third-Order Variable Step Size Superclass of Block Backward Differentiation Formula for Efficient Solution of Highly Stiff Differential Systems

Abstract

This paper develops a third-order fully implicit adaptive variable step-size superclass block backward differentiation formula (VSBBDF3) for the efficient numerical simulation of nonlinear stiff dynamical systems, including oscillatory, chaotic, and reaction-kinetics systems governed by ordinary differential equations. The method extends the classical block BDF framework through a structured superclass coefficient formulation while preserving full implicitness. By computing multiple solution approximations simultaneously within each block, the scheme enhances both stability and computational efficiency. An adaptive step-size strategy controls local truncation errors, enabling dynamic response to rapidly varying stiff behavior. Rigorous theoretical analysis establishes consistency, zero-stability, convergence, and A-stability, confirming their reliability for stiff problems. Nonlinear systems from the implicit formulation are efficiently handled using Newton-type iteration. Extensive numerical experiments on stiff linear, oscillatory, and nonlinear problems demonstrate that the proposed method consistently achieves higher accuracy with competitive computational cost compared to existing methods, including NBDF, VSBBDF, and MATLAB ODE solvers. The results further show that combining block formulation, superclass structure, and adaptive step-size control provides a more effective accuracy-efficiency balance. Overall, the VSBBDF3 method offers a robust, accurate, and efficient framework for the numerical simulation of nonlinear stiff dynamical systems, well suited for scientific and engineering applications in dynamics, control, and bifurcation analysis.