Eigenvalue Analysis of Spectral Discretisation Matrices from Legendre Dual Petrov-Galerkin Spectral Methods for IVPs

报告摘要：

         Eigenvalue distributions and behaviour of the corresponding eigen-matrix have much implication in the performance of a spectral algorithm. While the spectrum of second-order derivative operators for boundary value problems (BVPs) are well understood, the spectrum of spectral approximations of initial value problems (IVPs) are far under explored. In this talk, we shall show that the eigen-pairs of the spectral discretization matrices from the Legendre dual-Petrov-Galerkin spectral methods (LDPG) for IVPs are associated with the generalized Bessel polynomials. This allows us to precisely characterize the eigenvalue distributions and we then illustrate the applications of such findings in, e.g., theoretical foundation of time spectral methods, stability of explicit time discretisations of spectral methods for hyperbolic problems and parallel-in-time algorithms among others. Moreover, we shall also introduce well-designed space-time spectral methods for the study of interesting dynamics of the Kadomtsev-Petviashvili (KP) equation in oscillatory region. The contents of this talk will be mainly based on joint works with two CSC visiting PhD students: Desong Kong (Central South China University) and Xuping Wang (Southeast University of China).

报告人简介：

        Professor Wang Li-Lian is currently a full Professor of Applied Mathematics in the School of Physical and Mathematical Sciences in Nanyang Technological University (NTU), Singapore. Before joining NTU in 2006, he worked as a Postdoctoral Fellow and Visiting Assistant Professor at Purdue University in USA from 2002 to 2005. He received his PhD from Shanghai University in 2000 and then worked in Shanghai Normal University for two years.  His main research interest resides in spectral and high-order methods for PDEs. He has published about 100 papers in top scientific journals including Appl. Comput. Harmon. Anal., SIAM J. Numer. Anal., SIAM J. Appl. Math., Math. Comp., etc.. His co-authored Springer book on Spectral Methods (2011) has become a standard reference in this subject area.