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

Abstract

This study presents a recently developed class of approximate numerical integration techniques for solving first-order ordinary differential equations particularly 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 linear multistep methods performs competitively, and in several cases more effectively, than existing methods possessing similar structural and stability properties.