论文标题
非局部Cahn-Hilliard方程的二阶线性数值方案的双重稳定和收敛分析
Double stabilizations and convergence analysis of a second-order linear numerical scheme for the nonlocal Cahn-Hilliard equation
论文作者
论文摘要
在本文中，我们研究了非局部Cahn-Hilliard方程的二阶准确和线性数值方案。通过将修改的曲柄 - 尼科尔森近似值和Adams-bashforth外推以进行时间离散化来建立该方案，并通过将傅立叶光谱搭配应用于空间离散化。另外，为了数值稳定性，添加了不同形式的两个稳定项。我们通过使用数值方案的高阶一致性估计来进行完整的收敛分析，并结合粗糙的误差估计和精制估计。通过将数值解决方案作为精确解决方案的小扰动，我们能够证明由于粗糙误差估计值，数值解决方案的离散$ \ ell^\ infty $是合理的。随后，根据已建立的数值解决方案结合的$ \ ell^\ infty $，得出了精制的误差估计以获得最佳收敛速率。此外，相对于修改的能量，能量稳定性也得到了严格的证明。所提出的方案可以看作是早期工作中列出的二阶方案的概括，并且能量稳定性估计大大改善了其中的相应结果。
In this paper, we study a second-order accurate and linear numerical scheme for the nonlocal Cahn-Hilliard equation. The scheme is established by combining a modified Crank-Nicolson approximation and the Adams-Bashforth extrapolation for the temporal discretization, and by applying the Fourier spectral collocation to the spatial discretization. In addition, two stabilization terms in different forms are added for the sake of the numerical stability. We conduct a complete convergence analysis by using the higher-order consistency estimate for the numerical scheme, combined with the rough error estimate and the refined estimate. By regarding the numerical solution as a small perturbation of the exact solution, we are able to justify the discrete $\ell^\infty$ bound of the numerical solution, as a result of the rough error estimate. Subsequently, the refined error estimate is derived to obtain the optimal rate of convergence, following the established $\ell^\infty$ bound of the numerical solution. Moreover, the energy stability is also rigorously proved with respect to a modified energy. The proposed scheme can be viewed as the generalization of the second-order scheme presented in an earlier work, and the energy stability estimate has greatly improved the corresponding result therein.
