In this paper, we propose an efficient numerical scheme for the Allen–Cahn equations. We show theoretically using the Galerkin method and the compactness theorem that the solution of the afore-mentioned equation exists and is unique in appropriate spaces with the interaction length parameter $\alpha$ well controlled. We further, show numerically that the proposed scheme is stable and converge optimally in the $L^2$ as well as the $H^1$-norms with its numerical solution preserving all the qualitative properties of the exact solution. With the help of an example and a carefully chosen $\alpha$, we use numerical experiments to justify the validity of the proposed scheme.

中文翻译：

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使用 Galerkin 方法分析 Allen-Cahn 方程的有效数值格式

在本文中，我们为 Allen-Cahn 方程提出了一种有效的数值方案。我们使用伽辽金方法和紧性定理从理论上证明了上述方程的解存在并且在具有相互作用长度参数的适当空间中是唯一的$\alpha$控制得很好。我们进一步地，数值证明了所提出的方案是稳定的，并且在${升}^2$ 以及 $H^1$-norms 及其数值解保留精确解的所有定性属性。借助示例和精心挑选的$\alpha$，我们使用数值实验来证明所提出方案的有效性。