A Stiffly Stable Adaptive Fifth-Order Block Time-Stepping Method for Stiff Oscillatory Differential Equations

Abstract

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 be consistent, zero-stable, and achieves fifth-order accuracy. Its stability characteristics are further examined through A-stability analysis using locus boundary techniques, validating its applicability to stiff problems. Additionally, an error control procedure is incorporated to dynamically adjust the step size, ensuring an optimal balance between computational efficiency and solution accuracy. Numerical experiments on benchmark stiff oscillatory systems indicate that the proposed method outperforms the fourth-order variable step size block backward differentiation formula (VSBBDF4), as well as MATLAB solvers ODE15s and ODE23s, in terms of stability, accuracy, and computational efficiency.