Master Lecture: Academic Report by Professor Qin Sheng

Topic: A Call of Stability from a Multiscale Compact Scheme for Subwavelength Meta Optics Computation

Lecturer: Qin Sheng

Time: 10:00-12:00a.m., December 20, 2017 (Wednesday)

Venue: 145 Lecture Hall, School of Mathematics and Statistics

Abstract:

Rapid advances in subwavelength metal optics, e.g. nanophotonics, metamaterials, and plasmonics, have been demanding highly effective, efficient, and yet reliable PDE solvers. This is primarily due to broadband radiation absorptions of the metamaterials. Such optical properties are often tailorable from infrared to visible spectrums. Focusing features of the highly oscillatory beams through subwavelength metamaterials have been extremely difficult to calculate and simulate. For the simplicity, let us consider a radially symmetric electric field in transverse directions in this conversation. Thus, standard polar coordinates can be employed. To eliminate the transformation singularity 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. The 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. Can this multiscale algorithm still be vibrating in subwavelength Meta Optics applications? To this end, our investigation extends to a novel new definition of asymptotical stability. The original ideas of the consideration can be traced back to a 2007 research workshop together with Professor Shuhuang Xiang, CSU. In our study, highly oscillatory waves for subwavelength material applications are explored. Physical concerns are once 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. Computational experiments with laboratorial validations will be given to illustrate our conclusions.

Lecturer introduction:

Professor Qin Sheng is a Tenured Professor of Department of Mathematics and Department of Physics, US Baylor University. He is mainly engaged in the research on splitting of linear and nonlinear partial differential equations, adaptive algorithms and their applications, with research results known as Sheng-Suzuki theorem in the numerical analysis field. Professor Qin Sheng has achieved great academic attainments and carried out extensive cooperation with famous scholars around the world. Over the past 20 years, he has published more than 100 high quality papers, participated in the compilation of a number of academic monographs, and written “splitting algorithm” entry for Encyclopædia Britannica. Since 2010, Professor Qin Sheng has been acting as the Chief Editor of SCI journal - International Journal of Computer Mathematics. He has been repeatedly invited to visit universities and research institutions in the United States, Europe, Latin America and China, to attend international conferences on mathematics and deliver speeches, and has repeatedly gained the funding from US Department of Energy, US Department of Defense and National Natural Science Foundation of China. At present, he is instructor of 3 doctoral candidates and 1 postdoctoral researcher.