In this paper, we investigate and implement a numerical method that is based on the mimetic finite difference operator in order to solve the nonlinear Allen–Cahn equation with periodic and non-periodic boundary conditions. In addition, we also analyze the performance of this mimetic-based method by using the classical heat equation with a variety of boundary conditions. We assess the performance of the mimetic-based numerical method by comparing the errors of its solutions with those obtained by a classical finite difference method and the pdepde built-in Matlab function. We compute the errors by using the exact solutions when they are available or with reference solutions. We adapt and implement the mimetic-based numerical method by using the MOLE (Mimetic Operators Library Enhanced) library that includes some built-in functions that return representations of the curl, divergence and gradient operators, in order to deal with the Allen-Cahn and heat equations. We present several results with regard to errors and numerical convergence tests in order to provide insight into the accuracy of the mimetic-based numerical method. The results show that the numerical method based on the mimetic difference operator is a reliable method for solving the Allen–Cahn and heat equations with periodic and non-periodic boundary conditions. The numerical solutions generated by the mimetic-based method are relatively accurate. We also proposed a new method based on the mimetic finite difference operator and the convexity splitting approach to solve Allen-Cahn equation in 2D. We found that, for small time step sizes the solutions generated by the mimetic-based method are more accurate than the ones generated by the pdepe Matlab function and similar to the solutions given by a finite difference method.

中文翻译：

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使用模拟有限差分算子求解具有周期性和非周期性边界条件的 Allen-Cahn 方程


在本文中，我们研究并实现了一种基于模拟有限差分算子的数值方法，以求解具有周期性和非周期性边界条件的非线性 Allen-Cahn 方程。此外，我们还通过使用具有各种边界条件的经典热方程来分析这种基于模拟的方法的性能。我们通过将其解的误差与经典有限差分法和 pdepde 内置 Matlab 函数获得的误差进行比较，来评估基于模拟的数值方法的性能。我们通过使用精确解决方案（如果可用）或参考解决方案来计算误差。我们使用 MOLE （Mimetic Operators Library Enhanced） 库来调整和实现基于模拟的数值方法，该库包括一些返回旋度、发散和梯度运算符表示的内置函数，以处理 Allen-Cahn 和热方程。我们提出了有关误差和数值收敛测试的几个结果，以便深入了解基于模拟的数值方法的准确性。结果表明，基于模拟差分算子的数值方法是求解具有周期性和非周期性边界条件的 Allen-Cahn 和热方程的可靠方法。基于模拟的方法生成的数值解相对准确。我们还提出了一种基于模拟有限差分算子和凸分裂方法的新方法，用于求解二维 Allen-Cahn 方程。 我们发现，对于小时间步长，基于模拟的方法生成的解比 pdepe Matlab 函数生成的解更准确，并且类似于有限差分法给出的解。