Energy-stable EIEQ time discretization for the PFC–FCC model: Convergence analysis and applications to crystal growth

The phase-field crystal (PFC) model is widely used to describe atomic-scale crystalline patterns on diffusive time scales and to study the long-time evolution of microstructures. From the viewpoint of computational fluid dynamics, the standard two-mode PFC formulation for face-centered cubic ordering (PFC–FCC model) can be regarded as a high-order phase-field model for solid–liquid phase transitions, in which the motion of diffuse solid–liquid and grain boundaries is encoded in the evolution of a conserved order parameter. In this work, we constructed a linear, fully decoupled time-discretization scheme for the PFC-FCC system based on the explicit Invariant Energy Quadratization (EIEQ) approach. The scheme requires only the solution of constant-coefficient elliptic problems at each time step, making it well suited for fast solvers in large-scale simulations. We proved unique solvability, unconditional energy stability, and a rigorous optimal first-order a priori error estimate under suitable regularity assumptions, relying on a uniform $L^{\mathrm{\infty }}$-bound for the discrete density to control the nonlinear terms. Numerical tests with a manufactured solution confirmed the predicted convergence rate, while two- and three-dimensional simulations of solidification and crystallization with moving crystalline interfaces illustrated the stability, energy-dissipation property, and effectiveness of the EIEQ-based scheme as a numerical tool for long-time simulations of FCC/BCC pattern selection.