报告人：张智民（北京计算科学研究中心）

邀请人：李东方

报告时间：2022年5月12日 （星期四）9: 00-11:00  

报告地点：腾讯会议：135 997 961

报告题目：Efficient spectral methods and error analysis for nonlinear Hamiltonian systems

报告摘要：We investigate efficient numerical methods for nonlinear Hamiltonian systems. Three polynomial spectral methods (including spectral Galerkin, Petrov-Galerkin, and collocation methods). Our main results include the energy and symplectic structure preserving properties and error estimates. We prove that the spectral Petrov-Galerkin method preserves the energy exactly and both the spectral Gauss collocation and spectral Galerkin methods are energy conserving up to spectral accuracy. While it is well known that collocation at Gauss points preserves symplectic structure, we prove that the Petrov-Galerkin method preserves the symplectic structure up to a Gauss quadrature error and the spectral Galerkin method preserves the symplectic structure to spectral accuracy. Furthermore, we prove that all three methods converge exponentially (with respect to the polynomial degree) under sufficient regularity assumption. All these aforementioned properties make our methods possible to simulate the long time behavior of the Hamiltonian system. Numerical experiments indicate that our algorithms are efficient.

报告人简介：张智民，国际上知名的计算数学专家。他长期从事计算方法，尤其是有限元方法的研究，在超收敛、后验误差估计和自适应算法等领域的开拓性研究取得了多项创新成果。在国际上第一个建立起广为流行的ZZ离散重构格式的数学理论，并首次提出了基于多项式守恒的离散重构格式。发表SCI论文100余篇（包括国际顶尖计算数学杂志 SIAM Journal on Numerical Analysis、Mathematics of Computation、Numerische Mathematik)。