We consider fully discrete schemes based on the scalar auxiliary variable (SAV) approach and stabilized SAV approach in time and the Fourier-spectral method in space for the phase field crystal (PFC) equation. Unconditionally, energy stability is established for both first- and second-order fully discrete schemes. In addition to the stability, we also provide a rigorous error estimate which shows that our second-order in time with Fourier-spectral method in space converges with order O(Δt² + N^−m), where Δt, N, and m are time step size, number of Fourier modes in each direction, and regularity index in space, respectively. We also present numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of the schemes.

中文翻译：

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相场晶体方程的SAV傅里叶光谱方法的稳定性和误差估计

我们考虑基于时间的标量辅助变量（SAV）方法和稳定的SAV方法以及空间傅里叶谱方法（PFC）方程的完全离散方案。一阶和二阶完全离散方案都建立了无条件的能量稳定性。除了稳定性之外，我们还提供了严格的误差估计，该误差估计表明我们的傅里叶光谱方法在时间上的二阶收敛于O阶（Δt ² + N ^{- m}），其中Δt，N和米分别是时间步长，每个方向的傅立叶模式数和空间中的规律性指数。我们还提出了数值实验，以验证我们的理论结果，并证明了方案的鲁棒性和准确性。