论文标题
长时间$ \ MATHCAL H^S_α$具有多项式非线性的Cahn-Hilliard方程的经典方案的稳定性
Long time $\mathcal H^s_α$ stability of a classical scheme for Cahn-Hilliard equation with polynomial nonlinearity
论文作者
论文摘要
在本文中，我们研究了具有多项式非线性的Cahn-Hilliard方程的隐式Euler方案的长期稳定性。 $ h^{ - 1} $和$ \ MATHCAL H^S_α$（$ s = 1,2,3 $）的空间独立于初始数据和时间离散的步骤尺寸，用于该经典方案产生的数值解决方案，该方案具有可变的时间级别。借助于均匀的离散Gronwall引理得出。与连续到达动力学系统的估计值的比较表明，经典的隐式欧拉方法可以完全保留基础系统的长时间行为。借助傅立叶伪空间近似，数值实验也证明了如此长的行为。
In this paper we investigate the long time stability of the implicit Euler scheme for the Cahn-Hilliard equation with polynomial nonlinearity. The uniform estimates in $H^{-1}$ and $\mathcal H^s_α$ ($s=1,2,3$) spaces independent of the initial data and time discrete step-sizes are derived for the numerical solution produced by this classical scheme with variable time step-sizes.The uniform $\mathcal H^3_α$ bound is obtained on basis of the uniform $H^1$ estimate for the discrete chemical potential which is derived with the aid of the uniform discrete Gronwall lemma. A comparison with the estimates for the continuous-in-time dynamical system reveals that the classical implicit Euler method can completely preserve the long time behaviour of the underlying system. Such a long time behaviour is also demonstrated by the numerical experiments with the help of Fourier pseudospectral space approximation.
