This work discusses a way of allowing fast implicit update schemes for the temporal discretization of phase-field models for gradient flow problems that employ Fourier-spectral methods for their spatial discretization. Through the repeated application of the Sherman–Morrison formula we provide a rule for approximations of the inverted tangent matrix of the corresponding Newton–Raphson method with a selectable order. Since the representation of this inversion is exact for a sufficiently high approximation order, the proposed scheme is shown to provide a fixed-point-type iterative solver for gradient flow problems that require the solution of linear systems in the context of an implicit time-integration. While the proposed scheme is applicable to general gradient flow phase-field models, we discuss the scheme in the context of the Cahn–Hilliard equation, the Swift–Hohenberg equation, and the phase-field crystal equation for which we demonstrate the performance of the proposed method in comparison with classical solvers.

中文翻译：

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傅里叶谱方法中 Cahn-Hilliard 型梯度流的快速隐式更新方案


这项工作讨论了一种允许快速隐式更新方案的方法，用于梯度流问题的相场模型的时间离散化，该问题采用傅里叶谱方法进行空间离散化。通过谢尔曼-莫里森公式的重复应用，我们提供了相应牛顿-拉夫森方法的倒切矩阵的近似规则，并且具有可选择的阶数。由于该反演的表示对于足够高的近似阶数是精确的，因此所提出的方案为需要在隐式时间积分的背景下求解线性系统的梯度流问题提供了定点型迭代求解器。虽然所提出的方案适用于一般梯度流相场模型，但我们在 Cahn-Hilliard 方程、Swift-Hohenberg 方程和相场晶体方程的背景下讨论该方案，我们证明了该方案的性能所提出的方法与经典求解器进行比较。