Efficient spectral methods and error analysis for nonlinear Hamiltonian systems
报告人：张智民（北京计算科学研究中心&美国韦恩州立大学）
时间：2021-04-15 14:00-15:00
地点：腾讯会议
Abstract:
We investigate efficient numerical methods for nonlinear Hamiltonian systems. Three polynomial spectral methods: spectral Galerkin, Petrov-Galerkin, and collocation methods, are considered. 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.
主讲人简介：
张智民，中国科技大学学士（1982）、硕士（1985，导师石钟慈，中国科学院院士），美国马里兰大学博士（1991，导师Ivo Babuska，美国工程院院士）;美国韦恩州立大学(Wayne State University)教授、Charles H. Gershenson 杰出学者，北京计算科学研究中心讲座教授、应用和计算数学研究部主任，世界华人数学家大会两次45分钟报告人，现任和曾任10个国内外数学杂志编委，包括Mathematics of Computation、Journal of Scientific Computing、Numerical methods for Partial Differential Equations 、Journal of Computational Mathematics、Communications on Applied Mathematics and Computation、CSIAM Transaction on Applied Mathematics、《数学文化》等,发表SCI论文200余篇，主持过10个美国国家基金会的项目以及6个国家自然科学基金委的重点、面上、天元项目。
张智民教授长期从事计算方法，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,所提出的多项式保持重构（Polynomial Preserving Recovery—PPR）方法2008年被大型商业软件COMSOL Multiphysics 采用。
腾讯会议
https://meeting.tencent.com/s/sfab3QK5OXQH
会议 ID：960 715 783
会议密码：0415
