报告题目： Analysis of a linearized Crank-Nicolson scheme for the Schrödinger equation with logarithmic nonlinearity
报告人：闫静叶博士 （江苏大学）
时间：2022年6月23日(周四) 上午 9:30 -10:30
地点：藕舫楼802
主持人：王廷春教授
报告摘要：The logarithmic Schrödinger has a logarithmic nonlinearity f(u) = u ln |u|^2 that is not differentiable at u = 0. Compared with their counterparts with a regular nonlinear term, they possess richer and unusual dynamics, though the low regularity of the nonlinearity brings about significant challenges in both analysis and computation. Among very limited numerical studies, the semi-implicit regularized method via regularizing f(u) as u^εln(ε+ |u^ε|)^2 to overcome the blowup of ln |u|^2 at u = 0 has been investigated recently in literature. With the understanding of f(0) = 0, we propose and analyze a non-regularized semi-implicit (linearized) Crank-Nicolson scheme for these two types of logarithmic equations. The new tools for the error analysis include the characterization of the Holder continuity of the logarithmic term, and a nonlinear Gronwall’s inequality. Although there is an unpleasant (lnτ) ^2-factor in the exponential (which appears unavoidable as with the regularized method) of the error bounds, we indeed can recover the typical error estimates if the nonlinear term is Lipschitz continuous or more regular. We provide ample numerical results to demonstrate the expected convergence behaviours. We position this work as the first one to study the direct linearized scheme for such logarithmic equations as far as we can tell.
报告人简介：闫静叶，2020年12月博士毕业于国防科技大学，2019年8月至2020年8月赴新加坡国立大学，在包维柱教授指导下联合培养一年，2021年4月至2022年4月在新加坡南洋理工大学任研究助理。闫静叶博士主要从事偏微分方程数值解及其应用方面的研究，已在Journal of Computational Physics、Journal of Scientific Computing等高水平期刊发表学术论文10篇。
