Development of a least-squares-based polynomial partition of unity method for solving the 2D Allen–Cahn equation

Purpose This work presents a high-accuracy and computationally efficient numerical framework for solving the two-dimensional Allen–Cahn equation, a nonlinear phase-field model widely used to describe interface dynamics, phase separation and transport mechanisms relevant to heat and mass transfer processes. The proposed scheme employs a time-splitting strategy that effectively decouples the diffusion and reaction terms, enhancing numerical stability and allowing larger time steps without reducing accuracy. For spatial discretization, a polynomial-based partition of unity method is developed, in which locally-constructed least-squares polynomial approximations are smoothly blended to obtain a globally-accurate solution. Comprehensive numerical experiments demonstrate the robustness of the method and confirm its superior accuracy, stability and computational efficiency, underscoring its suitability for simulating phase-field evolution in heat and mass transfer applications. Design/methodology/approach A time-splitting numerical scheme is developed to solve the two-dimensional Allen–Cahn equation by decoupling the diffusion and reaction terms. The diffusion sub problem is treated using a polynomial-based partition of unity method, where local least-squares polynomial approximations are constructed and smoothly combined to form a global solution. The reaction term is handled separately in time, improving stability and allowing larger time steps. The proposed approach achieves high accuracy with reduced computational cost. Findings (1) A high-accuracy and computationally efficient numerical framework for solving the 2D Allen–Cahn equation is developed. (2) A time-splitting strategy is employed to decouple the diffusion and reaction terms, enhancing numerical stability and allowing for larger time steps. (3) A polynomial-based partition of unity method is proposed, combining locally-constructed least-squares polynomial approximations for a globally-accurate solution. (4) The method is particularly effective for simulating phase-field evolution in heat and mass transfer problems. (5) Extensive numerical experiments validate the robustness, accuracy and computational efficiency of the proposed method. Originality/value This study introduces a novel combination of a time-splitting strategy with a polynomial-based partition of unity framework for solving the Allen–Cahn equation. Unlike traditional discretization techniques, the proposed method integrates locally constructed least-squares polynomial approximations into a smooth global solution, achieving high accuracy with reduced computational effort. The approach provides an efficient and flexible alternative for phase-field simulations, offering improved stability and scalability for applications in heat and mass transfer.