报告题目：Stabilization parameter analysis of a second order linear scheme for the nonlocal Cahn-Hilliard equation

报 告 人：乔中华 教授

所在单位：香港理工大学

报告时间：2022年06月29日 星期三 15:30

报告地点：腾讯会议 ID：290-139-650

点击链接入会，或添加至会议列表：https://meeting.tencent.com/dm/54r1JuSrEi5A

校内联系人：张然 zhangran@jlu.edu.cn

报告摘要：A second order accurate (in time) and linear numerical scheme is proposed and analyzed for the nonlocal Cahn-Hilliard equation. The backward differentiation formula (BDF) is used as the temporal discretization, while an explicit extrapolation is applied to the nonlinear term and the concave expansive term. In addition, an $O (\dt^2)$ artificial regularization term, in the form of $A \Delta_N (\phi^{n+1} - 2 \phi^n + \phi^{n-1})$, is added for the sake of numerical stability. The resulting constant-coefficient linear scheme brings great numerical convenience;  however, its theoretical analysis turns out to be very challenging, due to the lack of higher order diffusion in the nonlocal model. In fact, a rough energy stability analysis can be derived, where an assumption on the $\ell^\infty$ bound of the numerical solution is required. To recover such an $\ell^\infty$ bound, an optimal rate convergence analysis has to be conducted, which combines a high order consistency analysis for the numerical system and the stability estimate for the error function. We adopt a novel test function for the error equation, so that a higher order temporal truncation error is derived to match the accuracy for discretizing the temporal derivative. Under the view that the numerical solution is actually a small perturbation of the exact solution, a uniform $\ell^\infty$ bound of the numerical solution can be obtained, by resorting to the error estimate under a moderate constraint of the time step size. Therefore, the result of the energy stability is restated with a new assumption on the stabilization parameter $A$. Some numerical experiments are carried out to display the behavior of the proposed second order scheme, including the convergence tests and long-time coarsening dynamics.

报告人简介：乔中华博士于2006年在香港浸会大学获得博士学位，现为香港理工大学应用数学系教授。

乔博士主要从事数值微分方程方面算法设计及分析，近年来研究工作集中在相场方程的数值模拟及计算流体力学的高效算法。他至今在SCI期刊上发表论文60余篇，文章被合计引用1300余次。他于2013年获香港研究资助局颁发2013至2014年度杰出青年学者奖，于2018年获得香港数学会青年学者奖，并且于2020年获得香港研究资助局研究学者称号。