An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation
An unconditionally stable hybrid numerical method for solving the Allen-Cahn equation
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DOI：
10.1016/j.camwa.2010.06.041
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发表时间：
2010-09
期刊：
影响因子：
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通讯作者：
Yibao Li;H. Lee;Darae Jeong;Junseok Kim
中科院分区：
文献类型：
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作者：
Yibao Li;H. Lee;Darae Jeong;Junseok Kim
我们提出了一种无条件稳定的二阶混合数值方法来求解代表二元混合物中反相区粗化模型的Allen-Cahn方程。该方法基于算子分裂技术。Allen-Cahn方程分为线性方程和非线性方程。首先，用Crank-Nicolson格式对线性方程进行离散，并用多重网格法等快速求解器求解离散后的方程组。然后，由于闭合形式解的可用性，对该非线性方程进行了解析求解。通过大量的数值实验，验证了该方法的准确性、有效性和稳定性。特别地，我们证明了该格式是无条件稳定的，在时间和空间上都是二阶精度的。
We present an unconditionally stable second-order hybrid numerical method for solving the Allen–Cahn equation representing a model for antiphase domain coarsening in a binary mixture. The proposed method is based on operator splitting techniques. The Allen–Cahn equation was divided into a linear and a nonlinear equation. First, the linear equation was discretized using a Crank–Nicolson scheme and the resulting discrete system of equations was solved by a fast solver such as a multigrid method. The nonlinear equation was then solved analytically due to the availability of a closed-form solution. Various numerical experiments are presented to confirm the accuracy, efficiency, and stability of the proposed method. In particular, we show that the scheme is unconditionally stable and second-order accurate in both time and space.