Space-Time Spectral Methods and Eigenvalue Analysis of Related Spectral Differentiation Matrices

主讲人：王立联 新加坡南洋理工大学教授

时间：2022年11月12日14：30

地点：腾讯会议 293 485 036

举办单位：数理学院

内容介绍：Spectral methods typically use global orthogonal polynomials/functions as basis functions which enjoy high-order accuracy and gain increasingly popularity in scientific and engineering computations. In most applications, spectral methods are employed in spatial discretizations but low-order schemes are used in time discretisations. This may create a mismatch of accuracy in particular for problems with evolving dynamics that require high-resolution in both space and time, e.g., oscillatory wave propagations. In this talk, we conduct eigenvalue analysis for the spectral discretization matrices for initial value problems based on the Legendre dual-Petrov-Galerkin spectral method (LDPG). While the spectrum of second-order derivative operators for boundary value problems are well understood, the spectrum of spectral approximations of initial value problems are far under explored. Here, we precisely characterise the eigen-pairs of the spectral discretisation matrices through the generalized Bessel polynomials. Such findings have much implication 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. We also introduce effective matrix decomposition algorithms to alleviate the burden of the extra works for spectral methods in time. This talk is based on joint works with Desong Kong (Central South China University), Jie Shen (Purdue University) and Shuhuang Xiang (CSU).