Abstract In this paper, we consider several splitting schemes for unsteady problems for the most common phase-field models. The fully implicit discretization of such problems would yield at each time step a nonlinear problem that involves second- or higher-order spatial operators. We derive new factorization schemes that linearize the equations and split the higher-order operators as a product of second-order operators that can be further split direction-wise. We prove the unconditional stability of the first-order schemes for the case of constant mobility. If the spatial discretization uses Cartesian grids, the most efficient schemes are Locally One Dimensional (LOD). We validate our theoretical analysis with 2D numerical examples.

中文翻译：

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相场模型的拆分方案

摘要 在本文中，我们考虑了最常见的相场模型的非定常问题的几种分裂方案。此类问题的完全隐式离散化将在每个时间步产生一个非线性问题，该问题涉及二阶或高阶空间算子。我们推导出新的分解方案，将方程线性化，并将高阶算子拆分为二阶算子的乘积，二阶算子可以进一步按方向拆分。我们证明了恒定迁移率情况下一阶方案的无条件稳定性。如果空间离散化使用笛卡尔网格，最有效的方案是局部一维 (LOD)。我们用二维数值例子验证了我们的理论分析。