A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation
A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation
复制标题
DOI：
10.4208/eajam.150713.181113a
复制
发表时间：
2013-11
影响因子：
1.2
通讯作者：
Lizhen Chen;Chuanju Xu;许传炬
中科院分区：
文献类型：
--
作者：
Lizhen Chen;Chuanju Xu;许传炬
本文提出并分析了一类具有Neumann边界条件的Cahn-Hilliard方程的全离散格式。该格式联合收割机了时间上的大时间步长分裂方法和空间上的谱元方法。我们特别感兴趣的是分析一类方法，分裂成低阶方程的原始Cahn-Hilliard方程。对于一阶分裂格式，基于能量方法研究了其稳定性和收敛性。证明了半离散和全离散解都满足相关连续问题中隐藏的能量耗散和质量守恒性质。一个严格的误差估计，连同数值验证，提供。虽然还没有严格证明，高阶格式也构造和测试的一系列数值例子。最后，将所提出的格式应用于复域中的相场仿真，得到了一些有趣的仿真结果。
We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard equation with Neumann boundary conditions. The schemes combine large-time step splitting methods in time and spectral element methods in space. We are particularly interested in analysing a class of methods that split the original Cahn-Hilliard equation into lower order equations. These lower order equations are simpler and less computationally expensive to treat. For the first-order splitting scheme, the stability and convergence properties are investigated based on an energy method. It is proven that both semi-discrete and fully discrete solutions satisfy the energy dissipation and mass conservation properties hidden in the associated continuous problem. A rigorous error estimate, together with numerical confirmation, is provided. Although not yet rigorously proven, higher-order schemes are also constructed and tested by a series of numerical examples. Finally, the proposed schemes are applied to the phase field simulation in a complex domain, and some interesting simulation results are obtained.