Look-Ahead Linear Multistep Methods for Initial Value Problems with Quadratic Stability Polynomials

Abstract

This study presents a recently developed class of approximate numerical integration 
techniques for solving first-order ordinary differential equations arising from initial value 
problems (IVPs). The proposed family of schemes, referred to as look-ahead linear multistep 
methods, is formulated based on the introduction of a function evaluation at an additional 
step beyond the current integration point. This look-ahead strategy enhances the 
computational structure of the methods and improves their numerical performance in the 
integration of a wide range of ordinary differential equations. One of the major advantages 
of the proposed approach is its ability to generate relatively large regions of absolute 
stability, particularly for methods whose stability polynomials possess quadratic 
characteristic roots. The methods further exhibit predictor–corrector characteristics, 
thereby improving convergence behavior, numerical consistency, and solution refinement 
during implementation. However, owing to the combined predictor–corrector formulation and 
the inherent look-ahead mechanism, the dependence of the stability roots on the independent 
variable also becomes quadratic in nature. In this work, the coefficients of the developed 
methods are derived explicitly up to order seven, while a systematic strategy for constructing 
higher-order schemes is also proposed. Preliminary numerical experiments conducted on 
selected initial value problems demonstrate that the developed methods are computationally 
efficient, highly accurate, and numerically reliable. Comparative analyses further reveal that 
the proposed class of look-ahead multistep methods performs competitively, and in several 
cases more effectively, than existing methods possessing similar structural and stability 
properties.