用于相场晶体方程的凸分裂谱方法：能量稳定性、计算稳定性图和三维GPU模拟
A Convex Splitting Spectral Method for the Phase Field Crystal Equation: Energy Stability, Computational Stability Maps, and Three-Dimensional GPU Simulations
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中文总结 AI 辅助
该研究针对相场晶体方程提出基于Eyre凸分裂框架的傅里叶谱方法，其格式无条件能量稳定且守恒质量。通过模拟得出经典条件\(a \geq 2\)保守，稳定区域内小\(a\)值误差更低，还验证了方法可扩展性，代码已开源。
中文摘要 AI 辅助
我们提出了一种基于Eyre凸分裂框架的高效傅里叶谱方法来求解相场晶体（PFC）方程。所提出的一阶格式无条件能量稳定且在机器精度上守恒质量。利用截断势论证建立了能量稳定性定理，从主导模能量平衡导出了半解析中性稳定性曲线，给出了实际稳定性边界的封闭形式表征。经典充分条件\(a \geq 2\)被证明是保守的：从40000次GPU加速的PFC模拟得到的计算稳定性图表明，\(a\)值显著低于该阈值时能量稳定解依然存在。关键的是，精度分析表明稳定区域内较小的\(a\)值始终产生更低的\(L^2\)误差。通过在\(256^3\)网格上进行长达\(T_f = 10000\)的长时间三维模拟的扩展渐近应力测试严格验证了这种高保真状态，成功执行\(10^6\)次连续时间增量，处于松弛稳定区域深处，远低于经典凸分裂极限（\(a < 2\)），同时保持严格单调的能量耗散和机器精度的质量守恒。最后，在单个消费级GPU上进行了分辨率高达\(512^3\)的二维和三维模拟，证明了所提出框架在无需HPC基础设施的情况下解决复杂相场动力学的可扩展性。代码已在GitHub上公开。
英文摘要
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 $a \geq 2$ 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.
