Eighth-Order Numerov-Type Methods Using Varying Step Length

This work explores a well-established eighth-algebraic-order numerical method belonging to the explicit Numerov-type family. To enhance its efficiency, we integrated a cost-effective algorithm for adjusting the step size. After each step, the algorithm either maintains the current step length, halves it, or doubles it. Any off-step points required by this technique are calculated using a local interpolation function. Numerical tests involving diverse problems demonstrate the significant efficiency improvements achieved through this approach. The method is particularly effective for solving differential equations with oscillatory behavior, showcasing its ability to maintain high accuracy with fewer function evaluations. This advancement is crucial for applications requiring precise solutions over long intervals, such as in physics and engineering. Additionally, the paper provides a comprehensive MATLAB-R2018a implementation, facilitating ease of use and further research in the field. By addressing both computational efficiency and accuracy, this study contributes a valuable tool for the numerical analysis community.