报告题目：On orthogonal systems and stable spectral methods for time-dependent PDEs

报告专家：Arieh Iserles教授（剑桥大学）

讲座时间：2022年5月17日（周二）17:00-18:00

腾讯会议ID：819 0338 3283    会议口令：123456

会议链接：https://us02web.zoom.us/j/81903383283?pwd=Vkc2SFFxSmdnNUo2S2hDMXB5K2RlZz09

Abstract:

Spectral methods are a powerful means to compute differential equations, yet they exhibit poor stability properties when applied to time-dependent problems. In this talk we argue that stability and, whenever required, energy conservation, follow once the differentiation matrix of an orthonormal system is (in a single space dimension) skew Hermitian and tridiagonal. This however imposes critical restriction on orthonormal systems complete in the Euclidean norm: essentially, they may exist only in three instances: the entire real line and a compact interval with either periodic or zero Dirichlet boundary conditions. The good news is that in all these cases skew-Hermiticity is attainable with a tridiagonal matrix. We completely characterise such systems: essentially, we establish a one-to-one connection between them and determinate Borel measures, and provide several examples. Further, we explicitly determine all such systems whose first n coefficients can be computed in O(n log n) operations. Finally, we debate the speed of convergence in different systems using both standard and asymptotic stability analysis.