In this work, we present a conservative Allen–Cahn (CAC) equation and investigate its unconditionally maximum principle-preserving linear numerical scheme. The operator splitting strategy is adopted to split the CAC model into a conventional AC equation and a mass correction equation. The standard finite difference method is used to discretize the equations in space. In the first step, the temporal discretization of the AC equation is performed by using the energy factorization technique. The discrete version of the maximum principle-preserving property for the AC equation is unconditionally satisfied. In the second step, we apply mass correction by using an explicit Euler-type approach. Without the constraint of time step, we estimate that the absolute value of the updated solution is bounded by 1. The unique solvability is analytically proved. In each time step, the proposed method is easy to implement because we only need to solve a linear elliptic type equation and then correct the solution in an explicit manner. Various computational experiments in two-dimensional and three-dimensional spaces are performed to confirm the performance of the proposed method. Moreover, the experiments also indicate that the proposed model can be used to simulate two-phase incompressible fluid flows.

中文翻译：

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具有局部拉格朗日乘子的质量守恒 Allen–Cahn 模型的无条件最大保原理线性方法


在这项工作中，我们提出了一个保守的 Allen–Cahn (CAC) 方程，并研究了其无条件最大保原理线性数值格式。采用算子分裂策略将CAC模型分裂为常规AC方程和质量修正方程。标准有限差分法用于在空间中离散方程。第一步，使用能量分解技术对 AC 方程进行时间离散。 AC 方程的最大保原理性的离散版本是无条件满足的。在第二步中，我们通过使用显式欧拉型方法来应用质量校正。在没有时间步长约束的情况下，我们估计更新解的绝对值以1为界。分析证明了唯一可解性。在每个时间步长中，所提出的方法很容易实现，因为我们只需要求解线性椭圆型方程，然后以显式方式校正解。在二维和三维空间中进行了各种计算实验，以证实所提出方法的性能。此外，实验还表明该模型可用于模拟两相不可压缩流体流动。