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

Lianghong Yuan; Haoyuan Wu; Yu Chen; Jun Zhang; Xiaofeng Yang; Lianghong Yuan; Haoyuan Wu; Yu Chen; Jun Zhang; Xiaofeng Yang

doi:10.3934/era.2026145

Electronic Research Archive

2026, Volume 34, Issue 5: 3223-3250. doi: 10.3934/era.2026145

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

1.
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China
2.
Moses Brown School, Providence, RI 02906, USA
3.
Graduate School of Business (GSB), SEGi University, Kota Damansara, Malaysia
4.
Computational Mathematics Research Center, Guizhou University of Finance and Economics, Guiyang 550025, China
5.
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA

Received: 16 January 2026 Revised: 25 February 2026 Accepted: 09 March 2026 Published: 14 April 2026

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^{\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.
- error estimate,
- EIEQ,
- fully-decoupled,
- first-order,
- PFC-FCC,
- energy stability
Citation: Lianghong Yuan, Haoyuan Wu, Yu Chen, Jun Zhang, Xiaofeng Yang. Energy-stable EIEQ time discretization for the PFC–FCC model: Convergence analysis and applications to crystal growth[J]. Electronic Research Archive, 2026, 34(5): 3223-3250. doi: 10.3934/era.2026145

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Abstract

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^{\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.

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