A Second-Order, Weakly Energy-Stable Pseudo-spectral Scheme for the Cahn–Hilliard Equation and Its Solution by the Homogeneous Linear Iteration Method
A Second-Order, Weakly Energy-Stable Pseudo-spectral Scheme for the Cahn–Hilliard Equation and Its Solution by the Homogeneous Linear Iteration Method
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DOI：
10.1007/s10915-016-0228-3
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发表时间：
2016-12
影响因子：
2.5
通讯作者：
Kelong Cheng;Cheng Wang;S. Wise;Xingye Yue
中科院分区：
文献类型：
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作者：
Kelong Cheng;Cheng Wang;S. Wise;Xingye Yue
本文对二维和三维Cahn-Hilliard方程提出了一个二阶能量稳定的数值格式，在空间上采用Fourier拟谱逼近。一个凸分裂处理保证了唯一的可解性和无条件能量稳定性的计划。同时，由于伪谱空间离散的全局性，非线性项的隐式处理使得直接非线性求解器不切实际。我们提出了一个齐次线性迭代算法来克服这个困难，其中（其中的时间步长）人工扩散项，道格拉斯-杜邦型正则化。因此，可以大大提高数值效率，因为高度非线性系统可以分解为纯线性求解器的迭代，这可以在伪谱设置中借助于FFT来实现。此外，一个仔细的非线性分析表明，这种线性迭代的压缩映射性质，在离散，离散Sobolev不等式应用。此外，还在理论水平上给出了数值解的范数上界。线性迭代求解器的效率证明在我们的数值实验。数值模拟结果显示了不同的能量衰减率值的卡恩-希利亚德流。
We present a second order energy stable numerical scheme for the two and three dimensional Cahn–Hilliard equation, with Fourier pseudo-spectral approximation in space. A convex splitting treatment assures the unique solvability and unconditional energy stability of the scheme. Meanwhile, the implicit treatment of the nonlinear term makes a direct nonlinear solver impractical, due to the global nature of the pseudo-spectral spatial discretization. We propose a homogeneous linear iteration algorithm to overcome this difficulty, in which an(wheresthe time step size) artificial diffusion term, a Douglas–Dupont-type regularization, is introduced. As a consequence, the numerical efficiency can be greatly improved, since the highly nonlinear system can be decomposed as an iteration of purely linear solvers, which can be implemented with the help of the FFT in a pseudo-spectral setting. Moreover, a careful nonlinear analysis shows a contraction mapping property of this linear iteration, in the discretenorm, with discrete Sobolev inequalities applied. Moreover, a bound of numerical solution innorm is also provided at a theoretical level. The efficiency of the linear iteration solver is demonstrated in our numerical experiments. Some numerical simulation results are presented, showing the energy decay rate for the Cahn–Hilliard flow with different values of.