非线性Hamiltonian 系统高效谱方法及其应用

The nonlinear Hamiltonian system is a dynamical system completely described by the Hamilton's equations and has many applications in classical mechanics, molecular dynamics, hydrodynamics, electrodynamics, plasma physics, relativity, astronomy, and other scientific fields..It is not easy to find their exact solutions for many nonlinear Hamiltonian systems describing practical engineering problems. On the contrary, an efficient approach is to find their numerical solutions. In addition, the nonlinear Hamiltonian system has some remarkable properties, most important among which are its symplectic structure and optimality for energy preservation. Any good numerical scheme should be able to replicate as many of these physical properties as possible. Thus, the project attempts to make a systematic and deep research on high effective numerical methods for nonlinear Hamiltonian system from three aspects: (1) To introduce proper Sobolev space and establish an efficient weak form and discrete scheme for nonlinear Hamiltonian system. (2) Propose an efficient iterative algorithm for solving the discrete scheme of nonlinear Hamiltonian system. （3）Apply the proposed numerical method to compute some practical problems, such as multi-body system, Henon-Heiles system, and so on.

非线性哈密顿系统是由哈密顿方程完全描述的动力系统，在经典力学、分子动力学、流体力学、电动力学、等离子体物理、相对论、天文学等领域有着广泛的应用。对于许多与实际工程问题相关的非线性哈密顿系统，很难找到其精确解。相反，一个有效的方法是找他们的数值解。另外，非线性哈密顿系统有一些显着的特性，其中最重要的是它的辛结构和保能量的最优性. 任何好的数值方法都应该尽可能多地保持这些物理特性。因此，本项目拟从三个方面对非线性哈密顿系统的高效数值计算方法作系统深入研究：(1) 引入适当的Sobolev空间，建立非线性哈密顿系统的一种有效的弱形式和离散格式。(2) 提出有效地求解离散的非线性Hamiltonian系统的一种迭代算法. （3）将所提出的数值方法用于一些实际问题的计算，如多体系统，Henon-Heiles系统等。