A Convex Splitting Spectral Method for the Phase Field Crystal Equation: Energy Stability, Computational Stability Maps, and Three-Dimensional GPU Simulations

Abstract

We present an efficient Fourier spectral method based on the convex splitting framework of Eyre~\cite{eyre1998unconditionally} for the phase field crystal (PFC) equation. The proposed first-order scheme is unconditionally energy stable and conserves mass to machine precision. An energy stability theorem is established using a truncated potential argument, and a semi-analytical neutral stability curve is derived from a dominant-mode energy balance, providing a closed-form characterization of the practical stability boundary. The classical sufficient condition $a\ge 2$ is shown to be conservative: a computational stability map obtained from 40,000 GPU-accelerated PFC simulations reveals that energy-stable solutions persist for values of $a$ significantly below this threshold. Crucially, accuracy analysis demonstrates that smaller values of $a$ within the stable region consistently yield lower $L^2$ errors. This high-fidelity regime is rigorously verified through an extended asymptotic stress test consisting of a long-time three-dimensional simulation on a $256^3$ grid up to $T_f=10,000$, successfully executing $10^6$ continuous temporal increments deep within the relaxed stability regime, well below the classical convex splitting limit ($a<2$), while preserving strict monotonic energy dissipation and machine-precision mass conservation. Finally, two-dimensional and three-dimensional simulations at resolutions up to $512^3$ are performed on a single consumer GPU, demonstrating the scalability of the proposed framework for resolving complex phase-field dynamics without requiring HPC infrastructure. The code is made publicly available on GitHub.

30 pages, and 22 figures