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首页> 外文期刊>Journal of Computational Physics >A weakly nonlinear, energy stable scheme for the strongly anisotropic Cahn-Hilliard equation and its convergence analysis
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A weakly nonlinear, energy stable scheme for the strongly anisotropic Cahn-Hilliard equation and its convergence analysis

机译：强烈各向异性CAHN-HALLIARD方程及其收敛分析的弱非线性，能量稳定方案

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摘要

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. (C) 2019 Elsevier Inc. All rights reserved.
机译：本文提出并分析了强烈各向异性CAHN-HALLIARD模型的弱非线性，能量稳定的数值方案。特别地，高度非线性和奇异的各向异性表面能使PDE系统在分析和数值水平上非常具有挑战性。为了克服这种众所周知的困难，我们对各向异性界面能量进行凸性分析，仔细估计揭示了所有其二阶函数衍生物的全局常数均匀。这种微妙的事实使得能够获得能量稳定的数值方案。此外，线性近似可用于表面能量部分，详细估计证明了相应的能量稳定性。它与非线性双重井电位术语的适当处理导致整个系统的弱非线性能量稳定方案。特别地，这种能量稳定性是关于相对于原始相变的界面能量而言，并且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原则的情况下。更重要的是，通过仔细应用全球界定的二阶功能衍生物，最佳速率收敛分析可用于所提出的数值方案，这是该区域的第一个这样的结果。同时，对于具有足够大的各向异性程度的CAHN-HILLIARD系统，必须引入一个WILLMORE或BIHAMOLOLOLONG规则化以使等式良好地提出。对于这种物理模型，所有所提出的分析仍然可用;可以以适当的方式导出独特的可解性，能量稳定性和收敛估计。另外，应用傅里叶伪光谱空间近似，并且可以为完全离散方案扩展所有理论结果。最后，提出了一些数值结果，这证实了所提出的方案的鲁棒性和准确性。 （c）2019 Elsevier Inc.保留所有权利。

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