Modeling and simulation of the conserved N-component Allen–Cahn model on evolving surfaces

Abstract

This paper establishes the conserved N-component Allen–Cahn model on evolving surfaces and conducts numerical simulations of the model. In mathematical modeling, since the surface motion velocity causes local contraction or expansion of the surface, it is hard to simultaneously fulfill the componential mass conservation and the point-wise mass conservation as the usual case on the static domain. Therefore, according to these two types of conservation, three models are established: the componential mass conservation model, the point-wise mass conservation model, and the componential and point-wise mass conservation model. For the numerical simulation, the evolving surface finite element method is used to discretize the model in time and space. To achieve a stable, linear, high-accuracy and decoupled numerical scheme, the evolving surface finite element method has been enhanced by incorporating the stabilized semi-implicit approach. Furthermore, the stability analysis has been undertaken to validate the robustness of the devised numerical scheme. Through the validation of numerous numerical simulations, the reasonableness of the proposed model and the efficacy of the numerical approach are evaluated.