Two Robust Computational Algorithms for a Few Nonlinear Roll Damping Models in Ocean Engineering: A Novel Graph Polynomial Approach

Abstract

Abstract. Accurate prediction of ship roll motion is challenging due to strong nonlinear damping effects that directly impact vessel safety. This study introduces two novel graph theoretic spectral algorithms, namely, the Stable Set Polynomial of the Complete Bipartite Graph Algorithm and the Clique Polynomial of the Complete Graph Algorithm for solving nonlinear ship roll motion equations with and without external excitation. The proposed methods transform the governing nonlinear differential equations into sparse algebraic systems using spectral polynomial representations over short time evolution intervals, enabling efficient and stable computation. Their performance is evaluated through systematic comparisons with the Homotopy Perturbation Method (HPM) and a standard high order explicit Runge–Kutta time integration scheme. To extend short time solutions to longer prediction horizons, a multilayer perceptron-based extrapolation strategy is employed. Numerical results, supported by root-mean-square error analyses and parameter space heatmaps, demonstrate improved accuracy and stability over existing methodologies, establishing the proposed algorithms as effective alternatives for nonlinear ship roll motion prediction.