Use of Multistep Methods for Solving ODEs

Abstract

Ordinary differential equations (ODEs) are used to model dynamic systems in many different fields of science and engineering. However, traditional multistep approaches like Adams–Bashforth and Adams–Moulton can fail when stiffness, nonlinearity, or great dimensionality. This research offers a framework that combines hybrid predictor–corrector methods with adaptive step sizing to increase accuracy and speed. We give Thorough stability and error analyses—including absolute stability regions and—are carried out using comparative benchmarks on representative standard test problems against conventional one-step and spectral solvers. Convergence characteristics. Applications range from stiff chemical kinetics, epidemic dynamics, climate inspired models, and nonlinear oscillators, therefore showing robustness under different situations. Numerical experiments show faster convergence, improved stability at larger time steps, and lower computational cost—reductions up to 40%—relative to standard approaches. The framework scales to contemporary large simulations, making hybrid multistep strategies an alternative to common solvers. We conclude with directions for further gains, including coupling machine learning with multistep integrators to drive dynamic step-size selection and predict error, stiffness, and stability margins. Overall, the results point to a practical, innovative path for advancing ODE solvers in sophisticated scientific computing where reliability, speed, and scalability are critical.