The localized exponential time differencing method based on overlapping domain decomposition has been recently introduced and successfully applied to parallel computations for extreme-scale numerical simulations of coarsening dynamics based on phase field models. In this paper, we focus on numerical solutions of a class of semilinear parabolic equations with the well-known Allen–Cahn equation as a special case. We first study the semi-discrete system under the standard central difference spatial discretization and prove the equivalence between the monodomain problem and the corresponding multidomain problem obtained by the Schwarz waveform relaxation iteration. Then we develop the fully discrete localized exponential time differencing schemes and, by establishing the maximum bound principle, prove the convergence of the fully discrete localized solutions to the exact semi-discrete solution and the convergence of the iterative solutions. Numerical experiments are carried out to verify the theoretical results in one-dimensional space and test the convergence and accuracy of the proposed algorithms with different numbers of subdomains in two-dimensional space.

中文翻译：

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基于重叠域分解的半线性抛物线方程指数时间差分方法

最近引入了基于重叠域分解的局部指数时间差分方法，并成功应用于基于相场模型的粗化动力学的极端尺度数值模拟的并行计算。在本文中，我们以著名的 Allen-Cahn 方程为特例，重点研究一类半线性抛物线方程的数值解。我们首先研究了标准中心差分空间离散化下的半离散系统，证明了单域问题与对应的 Schwarz 波形松弛迭代得到的多域问题之间的等价性。然后我们开发了完全离散的局部指数时间差分方案，并通过建立最大界原则，证明完全离散的局部解对精确半离散解的收敛性和迭代解的收敛性。数值实验验证了一维空间中的理论结果，并在二维空间中测试了所提出算法在不同子域数下的收敛性和准确性。