A New Methodology for the Development of Efficient Multistep Methods for First-Order IVPs with Oscillating Solutions

Abstract

In this research, we provide a novel approach to the development of effective numerical algorithms for the solution of first-order IVPs. In particular, we detail the fundamental theory behind the development of the aforementioned approaches and show how it can be applied to the Adams–Bashforth approach in three steps. The stability of the new scheme is also analyzed. We compared the performance of our novel algorithm to that of established approaches and found it to be superior. Numerical experiments confirmed that, in comparison to standard approaches to the numerical solution of Initial Value Problems (IVPs), including oscillating solutions, our approach is significantly more effective.

Efficient Multistep Algorithms for First-Order IVPs with Oscillating Solutions: II Implicit and Predictor–Corrector Algorithms

“…For implicit multistep approaches to first-order initial-value problems, we presented here the theory of phase-lag and amplification-error analysis. Several strategies for the construction of efficient predictor-corrector methods were offered, based on the theory described above and that developed in [115] for explicit methods. Our development efforts focused on the following strategies:…”

“…Recently, Simos [115] developed the theory for the development of multistep methods with minimal phase-lag or phase-fitted multistep methods for first-order IVPs. More specifically, he developed the theory for computing the phase-lag and amplification error of multistep methods for first-order IVPs.…”

“…It is worth mentioning that the idea put forward in this work and [115] is novel in the literature in relation to the development of:…”