In this article, we present a hybrid numerical scheme for solving the Allen–Cahn (AC) equation on a nonuniform mesh. The AC equation represents a model for antiphase domain coarsening in a binary mixture. To solve the AC equation on nonuniform grids, the AC equation is split into linear and nonlinear terms applying the operator splitting method. As the first step, the nonlinear term is solved using the separation of variables. Next, the diffusion term is decomposed into linear operators in each space direction. In each direction, we sequentially solve each diffusion equation by applying the implicit Euler method on a nonuniform grid to update the numerical solution. Because the implicit Euler method and analytic solution do not depend on the time step size, the proposed hybrid numerical method for the AC equation on a nonuniform mesh is unconditionally stable. In addition, we prove the proposed scheme satisfies the maximum principle. To verify the superior performance of the proposed method, we conduct numerical simulations such as motion by mean curvature, total energy non-increasing property, the maximum principle and unconditional stability.

中文翻译：

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非均匀网格上 Allen–Cahn 方程的混合数值方法

在本文中，我们提出了一种混合数值方案，用于求解非均匀网格上的 Allen–Cahn (AC) 方程。AC 方程表示二元混合物中反相域粗化的模型。为了求解非均匀网格上的 AC 方程，应用算子分裂方法将 AC 方程分为线性项和非线性项。第一步，使用变量分离来求解非线性项。接下来，扩散项被分解为每个空间方向上的线性算子。在每个方向上，我们通过在非均匀网格上应用隐式欧拉方法来更新数值解来顺序求解每个扩散方程。由于隐式欧拉方法和解析解不依赖于时间步长，因此所提出的非均匀网格上 AC 方程的混合数值方法是无条件稳定的。此外，我们证明了所提出的方案满足极大值原则。为了验证该方法的优越性能，我们进行了平均曲率运动、总能量不增性、极大值原理和无条件稳定性等数值模拟。