An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation
An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation
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DOI：
10.4208/cicp.300810.140411s
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发表时间：
2012-04
影响因子：
3.7
通讯作者：
Zhengru Zhang;Zhonghua Qiao
中科院分区：
文献类型：
--
作者：
Zhengru Zhang;Zhonghua Qiao
本文对描述相分离现象的Cahn-Hilliard方程进行了数值模拟。Cahn-Hilliard模型的数值模拟需要很长的时间才能达到稳态，因此大时间步长方法变得有用。这项工作的主要目的是构造无条件能量稳定的有限差分格式，以便在数值模拟中使用大时间步长。方程在空间上用中心差分格式离散，时间上用全隐式二阶格式离散。证明了该方案是无条件能量稳定和质量守恒的。给出了数值解在空间和时间上的二阶误差估计。在此基础上，提出了一种基于自由能随时间变化自适应选择时间步长的自适应时间步长策略。数值实验验证了自适应时间步长方法的有效性。
This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon. The numerical simulation of the Cahn-Hilliard model needs very long time to reach the steady state, and therefore large time-stepping methods become useful. The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations. The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time. The proposed scheme is proved to be unconditionally energy stable and mass-conservative. An error estimate for the numerical solution is also obtained with second order in both space and time. By using this energy stable scheme, an adaptive time-stepping strategy is proposed, which selects time steps adaptively based on the variation of the free energy against time. The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.