In this paper, we introduce an efficient numerical algorithm for solving the Lifshitz–Petrich equation on closed surfaces. The algorithm involves discretizing the surface with a triangular mesh, allowing for an explicit definition of the Laplace–Beltrami operator based on the neighborhood information of the mesh elements. To achieve second-order temporal accuracy, the backward differentiation formula scheme and the scalar auxiliary variable method are employed for Lifshitz–Petrich equation. The discrete system is subsequently solved using the biconjugate gradient stabilized method, with incomplete LU decomposition of the coefficient matrix serving as a preprocessor. The proposed algorithm is characterized by its simplicity in implementation and second-order precision in both spatial and temporal domains. Numerical experiments are conducted to validate the unconditional energy stability and efficacy of the algorithm.

中文翻译：

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一种具有无条件能量稳定性的二阶精确数值方法，用于曲面上的 Lifshitz-Petrich 方程


在本文中，我们介绍了一种在封闭表面上求解 Lifshitz-Petrich 方程的有效数值算法。该算法涉及使用三角形网格离散表面，从而允许根据网格单元的邻域信息显式定义 Laplace-Beltrami 算子。为了实现二阶时间精度，Lifshitz-Petrich 方程采用后向微分公式方案和标量辅助变量方法。随后使用双共购梯度稳定方法求解离散系统，系数矩阵的不完全 LU 分解用作预处理器。所提出的算法的特点是实现简单，在空间和时间域中都是二阶精度。通过数值实验验证了该算法的无条件能量稳定性和有效性。