A Hybrid Block–Pseudospectral Scheme for Spatiotemporal Pattern Formation in Phase-Field Models: The Allen–Cahn and Cahn–Hilliard Equations

This study introduces a hybrid numerical scheme for analysing spatiotemporal pattern formation in phase-field models, specifically the Allen–Cahn (AC) and the Cahn–Hilliard (CH) equations. These equations are important parabolic partial differential equations (PDEs) that arise in many interesting scientific problems. However, their strong nonlinearity makes finding analytical solutions practically impossible, while traditional numerical methods often suffer from instability. To address this challenge, this study proposes a robust hybrid block–pseudospectral method (HB–PSM) that integrates the off-step hybrid block method (HBM) for temporal integration, Chebyshev pseudospectral for spatial discretisation, and the quasilinearisation method (QLM) to handle nonlinearity. The primary aim is to develop a high-order, energy-stable numerical solver that accurately captures spatiotemporal patterns in both 1D and 2D configurations at reasonable computational cost. Through extensive numerical experimentation, the findings demonstrate that the HB–PSM is a competitive method for solving gradient flow systems, and it can be applied more broadly across many fields to solve a broader class of complex, multidimensional partial differential equations.