This paper presents an adaptive block time-stepping algorithm designed for the numerical integration of first-order stiff oscillatory problems, which typically involve multiple time scales and stringent stability conditions. The proposed method integrates a block formulation enabling the simultaneous evaluation of several solution points with an adaptive step size mechanism to effectively regulate local truncation errors. A predictor-corrector strategy is employed, where an explicit predictor generates initial estimates, and an implicit block corrector enhances stability for stiff systems. The resulting nonlinear equations are resolved using Newton's iterative method. The scheme is proven to... More >

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 respons... More >