報告題目：Stabilization parameter analysis of a second order linear scheme for the nonlocal Cahn-Hilliard equation
報 告 人：喬中華 教授
所在單位：香港理工大學
報告時間：2022年06月29日 星期三 15:30
報告地點：騰訊會議 ID：290-139-650
點擊鍊接入會，或添加至會議列表：https://meeting.tencent.com/dm/54r1JuSrEi5A
校内聯系人：張然 zhangran@jlu.edu.cn
報告摘要：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.
報告人簡介：喬中華博士于2006年在香港浸會大學獲得博士學位，現為香港理工大學應用數學系教授。
喬博士主要從事數值微分方程方面算法設計及分析，近年來研究工作集中在相場方程的數值模拟及計算流體力學的高效算法。他至今在SCI期刊上發表論文60餘篇，文章被合計引用1300餘次。他于2013年獲香港研究資助局頒發2013至2014年度傑出青年學者獎，于2018年獲得香港數學會青年學者獎，并且于2020年獲得香港研究資助局研究學者稱号。
