报 告 人：吴奕飞
报告时间：3月24日16:00
报告地点：位育楼217
报告题目：Second-order uniformly accurate method for the semiclassical nonlinear Schr\"odinger equation in $H^2$
报告摘要： In this talk, we consider the numerical solution of the semiclassical nonlinear Schr\"odinger equation on the d-dimensional torus $\mathbb{T}^d$, with highly oscillatory initial data depending on a small parameter $\varepsilon \in (0,1]$. We first show that a WKB-type approximation attains an $\mathcal{O}(\varepsilon)$ error in the $L^2$ norm for $H^2$ initial data theoretically, although its accuracy deteriorates as $\varepsilon$ increases. To address this limitation, we propose a numerical scheme that (i) applies a Galilean transform to remove the oscillations in the initial data, (ii) establishes sharp space--time estimates for the transformed equation, and (iii) employs a new low-regularity integrator to achieve second-order accuracy under the minimal $H^2$ regularity, which is weaker than the regularity assumptions in the literature. Furthermore, our analysis shows that the CFL-type conditions linking $h$, $\tau$, and $\varepsilon$ --- typically imposed in the semiclassical regime in the literature --- are not required in our scheme to obtain second-order convergence with respect to $\tau$ and $h$, uniformly with respect to $\varepsilon$, under the weaker regularity condition. Numerical experiments support the theoretical results and demonstrate the robustness of the method across a wide range of $\varepsilon$.
报告人简介：吴奕飞，南京师范大学数学科学学院院长，教授，博士生导师，国家级领军人才。主要从事偏微分方程理论及数值分析，调和分析等方向的交叉研究工作，在非线性Schrodinger方程，KdV方程等整体适定性和低正则算法构造方面做出一系列研究成果，在JEMS, Mem. AMS，Found. Comput. Math.，CMP，Adv. Math.， Anal. PDE，Numer. Math，Math. Comp, SINUM等期刊发表论文。