无穷维哈密尔顿系统保结构算法的研究与应用

有限维哈密尔顿系统的保结构算法已经日趋完善，广泛应用于分子动力学、天体轨道等领域。而非线性波的传播，电磁场的演化等实际问题经常涉及无穷维哈密尔顿系统。对该系统的保结构算法研究处于起步阶段，仍有众多基础理论与实际应用问题函待解决。本项目着重从如下两方面深入研究无穷维哈密尔顿系统的保结构算法。一是继续发展无穷维哈密尔顿系统保结构算法的基本理论，重点考虑如下三方面内容：非周期边界条件下保结构算法的构造和分析；高阶局部保结构算法的构造、分析和快速求解，重点构造和分析高效和高精度局部保能量算法；有限元和谱Galerkin方法的保结构性质研究，以及基于这两种方法构造的多辛算法和能量守恒算法。二是研究新构造的保结构算法在非线性波方程和电磁波方程数值模拟上的应用。本项目不仅可以从理论上进一步完善无穷维哈密尔顿系统保结构算法，还将为一些实际问题给出一系列高效、稳定的保结构算法以及有价值的数值模拟结果。

Structure-preserving algorithms for finite dimensional Hamiltonian systems are being perfected, and have been widely applied to the study of molecular dynamics, planetary orbits, and so on. However, the realistic problems such as the propagation of nonlinear waves, the evolution of electromagnetic fields often involve infinite dimensional Hamiltonian systems. The structure-preserving algorithms for such systems are still in the early stage, where there are many issues about fundamental theories and practical applications waiting to be solved. Therefore, this project further investigates the structure-preserving algorithms for infinite dimensional Hamiltonian systems wtih the following two parts. In the first part, we develop continuously the basic theory of structure-preserving algorithms for infinite dimensional Hamiltonian systems. There are three contents in the first part. The first one is the construction and analysis of structure-preserving algorithms for the equations under non-periodic boundary conditions. The second one is the construction and analysis of high-order local structure-preserving algorithms and their fast solvers. Here the emphasis is taken on the local energy-preserving algorithms. The third one is the investigation of structure-preserving properties of finite element methods and spectral Galerkin methods, as well as the associated multisymplectic algorithms and energy-preserving algorithms. The second part is for the applications of the new constructed algorithms to the simulations of nonlinear waves and electromagnetic waves. This project can not only improve the basic theory of the structure-preserving algorithms for infinite dimensional Hamiltonian systems, but also propose series of efficient and stable structure-preserving algorithms, as well as some valuable numerical results, for some problems in practical fields.