Mean-Driven Accuracy: A Contraharmonic Extension of Heun’s Method

This paper presents a novel modification of Heun’s method for solving initial value problems in ordinary differential equations (ODEs) by incorporating contraharmonic and centroidal means into the corrector step. Unlike traditional Heun’s method, which relies on the arithmetic means to average the slopes, our approach generalizes this averaging using nonlinear means that better capture the curvature and dynamics of the solution trajectory. Numerical simulations demonstrate that the proposed method offers improved accuracy and enhanced stability, particularly in stiff or rapidly changing systems. Applications include Newton’s law of cooling under standard and extreme thermal conditions, where our method consistently maintains accuracy and robustness. The results suggest that contraharmonic and centroidal means provide a viable and efficient alternative to conventional averaging strategies in explicit predictor–corrector methods.