A weakly nonlinear, energy stable scheme for the strongly anisotropic Cahn-Hilliard equation and its convergence analysis
A weakly nonlinear, energy stable scheme for the strongly anisotropic Cahn-Hilliard equation and its convergence analysis
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DOI：
10.1016/j.jcp.2019.109109
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发表时间：
2020-03
期刊：
影响因子：
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通讯作者：
Kelong Cheng;Cheng Wang;S. Wise
中科院分区：
文献类型：
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作者：
Kelong Cheng;Cheng Wang;S. Wise
本文提出并分析了强各向异性Cahn-Hilliard模型的弱非线性能量稳定数值格式。特别是，高度非线性和奇异各向异性的表面能使得PDE系统在分析和数值层面上都非常具有挑战性。为了克服这一众所周知的困难，我们对各向异性界面能进行了凸性分析，仔细估计表明它的所有二阶泛函导数都一致地由一个全局常数限定。这个微妙的事实使我们能够推导出一个能量稳定的数值格式。此外，表面能部分可以得到线性近似，详细的估计表明了相应的能量稳定性。结合对非线性双井势项的适当处理，得到了整个系统的弱非线性能量稳定方案。特别地，这种能量稳定性是以界面能相对于原始相变量来表示的，不需要引入辅助变量。这具有重要的含义，例如，在方法需要满足最大值原则的情况下。更重要的是，通过对二阶泛函导数的全局界的仔细应用，可以对所提出的数值格式进行最优速率收敛分析，这是该领域的第一个这样的结果。同时，对于具有足够大的各向异性的Cahn-Hilliard系统，必须引入Willmore或双调和正则化来使方程适定。对于这样一个物理模型，所有提出的分析仍然可用；通过适当的方法可以得到唯一的可解性、能量稳定性和收敛性估计。此外，应用傅里叶伪谱空间逼近，所有理论结果都可以推广到完全离散格式。最后给出了一些数值结果，验证了该方法的鲁棒性和准确性。
In this paper we propose and analyze a weakly nonlinear, energy stable numerical scheme for the strongly anisotropic Cahn-Hilliard model. In particular, a highly nonlinear and singular anisotropic surface energy makes the PDE system very challenging at both the analytical and numerical levels. To overcome this well-known difficulty, we perform a convexity analysis on the anisotropic interfacial energy, and a careful estimate reveals that all its second order functional derivatives stay uniformly bounded by a global constant. This subtle fact enables one to derive an energy stable numerical scheme. Moreover, a linear approximation becomes available for the surface energy part, and a detailed estimate demonstrates the corresponding energy stability. Its combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear, energy stable scheme for the whole system. In particular, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, and no auxiliary variable needs to be introduced. This has important implications, for example, in the case that the method needs to satisfy a maximum principle. More importantly, with a careful application of the global bound for the second order functional derivatives, an optimal rate convergence analysis becomes available for the proposed numerical scheme, which is the first such result in this area. Meanwhile, for a Cahn-Hilliard system with a sufficiently large degree of anisotropy, a Willmore or biharmonic regularization has to be introduced to make the equation well-posed. For such a physical model, all the presented analyses are still available; the unique solvability, energy stability and convergence estimate can be derived in an appropriate manner. In addition, the Fourier pseudo-spectral spatial approximation is applied, and all the theoretical results could be extended for the fully discrete scheme. Finally, a few numerical results are presented, which confirm the robustness and accuracy of the proposed scheme.