Spectral and spectral collocation methods for Hamiltonian systems

本项目的目标是发展一些稳健的、高精度的哈密顿系统的数值算法和相关的数学理论。研究工作将致力于构建“本质上”(精度在计算机舍入误差范围内)体积守恒(辛算法)和能量守恒算法。该项目将使用谱方法、有限元超收敛以及间断Galerkin方法中的一些最新数学理论。经典力学、分子动力学、流体力学、电动力学、等离子体物理、相对论和天文学等不同学科中的现象可以用哈密顿动力学系统来描述。该项目的成功将对科学和工程实践产生影响。这项研究将拓宽科学界在数学理论和实用算法设计方面的知识体系。该项目包含了坚实的多学科和跨学科的组成部分，在科学计算中有广泛的应用。

The objective of this project is to develop some robust and high accuracy numerical algorithms and related mathematical theory for Hamiltonian systems. The research efforts will be devoted to constructing "essentially" (accuracy is in a range of the computer round-off error) volume conserving (symplectic) and energy conserving algorithms. Some recent mathematical theory in spectral methods, finite element superconvergence, as well as discontinuous Galerkin methods will be employed in the project.Phenomena in different scientific disciplines such as classical mechanics, molecular dynamics, hydrodynamics, electrodynamics, plasma physics, relativity, and astronomy, etc. can be described by Hamiltonian dynamical systems. The success of the project will impact science and engineering practices. This research will widen the body of knowledge in the scientific community on both mathematical theory and practical algorithmic design. The project contains solid multi- and interdisciplinary components and has wide application in scientific computing.