题目：A fully discrete finite difference scheme for the  Flory-Huggins-Cahn-Hilliard equation with dynamical  boundary conditions

报告人:王成教授（美国马塞诸塞州立大学达特茅斯分校）

时间：2022年5月12日（周四）10:00-11:00

线上报告：Zoom网址：https://umassd.zoom.us/j/6339263022 密码：123456

摘 要 ： Abstract: A fully discrete finite difference numerical scheme is proposed and analyzed for the Cahn-Hilliard equation with a logarithmic Flory Huggins energy potential, combined with the dynamic boundary condition. Such a boundary condition couples the interior evolution of the gradient flow with the boundary evolution, and a dissipation law becomes available for the total energy, including the bulk energy and surface energy. The centered finite difference spatial approximation is taken. In the temporal discretization, we treat the nonlinear logarithmic terms and the surface diffusion term implicitly, and update the linear expansive term explicitly. Such a convex splitting approach is applied at both the interior dynamics and the boundary evolution. In turn, a careful calculation reveals that, the implicit part of the numerical system corresponds to a minimization of strictly convex discrete energy functional. In particular, the coefficient for the singular logarithmic terms becomes $( 1 + 2 h^{-1} )$ on the boundary points, in comparison with the regular coefficient, given by $1$, at interior grid points. This subtle fact leads to the well-posed feature of the numerical system. A theoretical justification of the unique solvability and positivity-preserving property of this numerical algorithm is provided, so that the phase variable is always between $-1$ and 1 at a point-wise level. This analysis is based on the following fact: the singular nature of the logarithmic term around the values of $-1$ and 1 prevents the numerical solution reaching these singular values, so that the numerical scheme is always well-defined as long as the numerical solution stays bounded at the previous time step. In addition, an unconditional energy stability of the numerical scheme is derived, without any restriction for the time step size. The optimal rate convergence analysis is presented as well.

个人简介：王成，1993年毕业于中国科技大学获数学学士学位，2000年在美国坦普尔大学获得博士学位,。2000-2003年在美国印尼安纳大学做博士后，2003-2008年在美国田纳西大学任助理教授，2008-2012年在美国麻省大学达特茅斯分校任助理教授，2012年晋升为副教授，2019年晋升为教授。主要研究领域是应用数学，包括数值分析、偏微分方程、流体力学、计算电磁学等。在Journal of Computational Physics，SIAM Journal on Numerical Analysis，IMA Journal of Numerical Analysis等期刊上发表论文五十多篇。