1、 1 2018年 南京农业大学 理学院数学学术 交流系列报告 （一） - 微分方程保结构算法 及应用 时间： 2018 年 3 月 7 日（星期三）下 午 3:00 6:00 地点：理学院教四楼 B211 1. 名师 Qin Sheng 教授 （美国 Baylor大学） 学术报告 A Multiscale Compact Scheme for Subwavelength Applications: Is it Numerically Stable? Abstract： Rapid advances in nanophotonics constantly demand highly effect
2、ive, efficient, and yet reliable PDE solvers. This is primarily due to broadband radiation absorptions of the metamaterial models which can be tailored from infrared to visible spectrums which have important applications in not only engineering but also biological pathways. Focusing features of the
3、oscillatory waves through subwavelength gaps have been extremely difficult to calculate and simulate. For the simplicity, let us consider a radially symmetric field in transverse directions in this conversation. Thus, standard polar coordinates can thus be employed. To eliminate the transformation s
4、ingularity occurred, we deploy a transverse domain decomposition which enables a multiscaled environmental setting that allows multi-feature wave approximations. We then consider a multiscaled compact method for a paraxial Helmholtz equation modeling nanobeams focusing through subwavelength holes. T
5、he compound numerical method is straightforward, simple-to-use. However, we can show that such a highly accurate compact scheme shies away from the stability in the conventional von Neumann sense. But, can this multiscale algorithm still be vibrating in modeling applications? To answer, our investig
6、ation extends to a novel new concept of asymptotical stability. The original ideas of the study come from a famous research workshop in the Isaac Newton Institute, 2007. In our study, highly oscillatory migration waves for subwavelength material applications are explored. Physical concerns are once
7、again placed before traditional mathematical arguments. Intensive auxiliary expansions and analysis are carried out. It is proven that, while appropriate constraints are reinforced, the asymptotical stability of aforementioned multiscale compact method remains affective. Computer experiments with la
8、boratorial validations will be given to illustrate our conclusions. 报告人简介： Qin Sheng 教授是美国贝勒大学数学系和物理系终身教授。 Qin Sheng 教授的主要研 究2 领域为线性和非线性偏微分方程的分裂与自适应算法及其应用，其主要研究结果在数值分析领域被誉为 Sheng-Suzuki 定理。 Qin Sheng 教授学术造诣极深，与世界各地的知名学者有这广泛深入的合作。在二十多年的研究生涯中， Sheng 教授发表了超过 100 篇高质量论文，参与编写多部学术专著，并且为大英百科词典撰写分裂算法的词条。 20
9、10 年起， Qin Sheng 教授担任国际 SCI 期刊 International Journal of Computer Mathematics 主编。 Qin Sheng 教授多次应邀访问美国、欧洲、拉丁美洲和中国的高校与研究机构，多次应邀参加国际数学大会并做邀请报告，多次获得美国能源部、国防部以及国家自然科学基金委的资助。目前， Qin Sheng 教授指导 3 名博士研究生和 1 名博士后研究人员。 2. 杨红莉 博士 、曾 宪 阳博士（南京工程学院） Construction of energy-preserving/dissipative continuous Runge-K
10、utta methods Abstract： The aim of this talk is to systematically study the structure of the energy-preserving/dissipative continuous RK method (EP/D-cRK method). The EP/D-cRK method can preserve the energy conservation of the Hamiltonian system, and can also preserve the energy dissipation of the Ly
11、apunov systems. In this talk, the energy conservation and dissipation theorem of continuous Runge-Kutta method (cRK method) is established on the basis of the research status of cRK method. Based on this theorem, it is found that there is only one 1-stage EP/D-cRKmethod, and that for any s-stage EP/
12、D-cRK method it is at least 2nd order. At last one new 2-stage 3rd order EP/D-cRK method is given. Numerical experiments show that it has good numerical behavior. 报告人简介： 杨红莉， 女， 南京工程学院数理部讲师 ， 2009年毕业于 南京大学 理学博士。 2008年获国家留学基金委资助， 访问德国图宾根大学 1年 。长期 从事常微分方程数值解的研究工作， 主持江苏省自然科学基金青年基金项目 1项及国家自然科学基金青年基金项目 1
13、项。 主要 研究兴趣 涉及振荡微分方程、 Hamilton系统、 RK型方法、保结构方法和有根树理论等，部分研究 成 果 收录在 Springer出版社的专著 Structure-Preserving Algorithms for Oscillatory Differential Equations I的第三章及 Structure-Preserving Algorithms for Oscillatory Differential Equations II的第十一章 。目前已发表论文十多篇，其中SCI收录论文 5篇，授权的发明专利 5项，实用新型专利 7项。 3. 刘 凯 博士（南京财经大学
14、） 、石玮博士（南京工业 大学） High-order skew-symmetric differentiation matrix on symmetric grid Abstract: E. Hairer and A. Iserles, “Numerical stability in the presence of variable coefficients“, Found Comput. Math, 2016 presented a detailed study of skew-symmetric matrix approximation to a first derivative whi
15、ch is proved to be fundamental in ensuring stability of discretisation for evolutional partial differential equations with variable coefficients. An open problem is proposed in that paper which concerns about the existence and construction of the perturbed grid that supports high-order skew-symmetri
16、c differentiation matrix for a given grid and only the case p=2 for this problem have been solved. This paper is an attempt to solve the problem for any p2. We focus ourselves on the symmetric grid and prove the existence of the perturbed grid for 3 arbitrarily high order p and give in detail the co
17、nstruction of the perturbed grid. Numerical experiments are carried out to illustrate our theory. 报告人简介： 刘凯，南京财经大学讲师， 2015年毕业于南京大学数学系，获理学博士学位。 2014年获国家留学基金委资助，访问剑桥大学 数学中心 Arieh Iserles 教授 1年 。 主要研究领域为 微分方程保结构算法 ， 在 J. Comput. Math.， Numer. Algor. , Comput. Phys. Commun.等期刊发表 SCI论文 10余篇，并与博士导师合作编写出版学
18、术专著 1本 Structure-Preserving Algorithms for Oscillatory Differential Equations II（ Springer） 。目前主持国家自然科学基金青年基金项目 1项。 4. 刘 长迎 博士（南京 信息工程 大学） Nonlinear stability and global error analysis of ERKN integrators for multi-frequency highly oscillatory systems with applications to semi-linear wave equations
19、with a takanami number Abstract: In this paper, we first investigate the nonlinear stability and convergence of the extended Runge-Kutta-Nystr“om (ERKN) integrators for multi-frequency highly oscillatory systems $q(t)+kappa2Aq(t)=gbig(q(t)big)$ with a takanami number $kappa2$ and a dominant frequenc
20、y-matrix $A$. Then the global errors of the ERKN-type time integrators are rigorously analysed when applied to semi-linear wave equations. Based on the finite-energy condition, the theoretical results turn out that the nonlinear stability and the global error bounds are entirely independent of the t
21、akanami number $kappa2$ and the dominant frequency-matrix $A$, or the spatial differential matrix deduced from a suitable spatial discretisation for the underlying semi-linear wave equation. The analysis also provides a new perspective on the ERKN-type integrators. That is, the ERKN-type time integr
22、ators are free from the restriction on the CFL condition when applied to semi-linear wave equations. This is an essential attribute of ERKN-type integrators which plays a key role. Unfortunately, however, the CFL condition is required for the vast majority of traditional schemes when applied to semi
23、-linear wave equations. Numerical experiments verify our theoretical analysis results and demonstrate the remarkable superiority of the ERKN-type time integrators in comparison with the traditional numerical approaches in the literature. 报 告人 简 介： 刘长迎， 南京信息工程大学讲师， 2013年于东南大学 获 理学硕士学位， 2017年在南京大学获 博士学位， 2015年 3月 -2015年 9月获国家留学基金委资助，访问剑桥大学数学中心 Arieh Iserles 教授 。主要研究兴趣：微分方程的几何数值积分、发展 微分方程的保结构算法、 Hamilton 系统及 Lyapunov 系统的高阶保能量方法，已在 国际权威期刊 发表 11篇 SCI论文。
