哈密尔顿系统高效高精度算法研究及其应用

哈密尔顿系统是一类非常重要的动力系统，它在经典力学、分子动力学、流体力学等科学领域有着广泛的应用。由于精确求解非线性哈密尔顿系统通常是比较困难的，寻求高效高精度的数值方法是很多学者一直关注的问题。因此，本项目拟从三个方面提出非线性哈密尔顿系统的一些新的高效高精度数值方法：（1）提出非线性哈密尔顿系统基于勒让德多项式的一种高阶谱变分积分子方法，从而以较少的自由度获得具有谱精度的数值解。并对该算法的保辛性质，非线性迭代的收敛性等性质从理论上加以分析和证明。（2）提出非线性哈密尔顿系统基于SAV格式和区域分解的一种有效的谱Petrov-Galerkin方法。并对该算法的保能量性质，数值解的收敛性等性质从理论上加以分析和证明。（3）在上面两种算法的基础上，提出量子非线性动力学方程组的一种有效的谱方法，并将该算法用于研究非线性效应对量子系统的影响。

Hamiltonian system is a kind of very important dynamical system, which is widely used in classical mechanics, molecular dynamics, hydrodynamics and other scientific fields. Because it is usually difficult to solve the nonlinear Hamiltonian system analytically, many scholars have been paying attention to the problem of finding efficient and high-precision numerical methods. Thus, this project attempts to propose some new efficient and high-precision numerical methods for nonlinear Hamiltonian system from three aspects: （1） Propose a high order spectral variational integrators method based on Legendre polynomials for nonlinear Hamiltonian system, and obtain the numerical solution with spectral accuracy with less degrees of freedom. Further theoretically analyze and prove the symplectic preserving property and the convergence of nonlinear iteration. (2) Propose an effective spectral Petrov-Galerkin method based on SAV scheme and domain decomposition for nonlinear Hamiltonian systems, and further theoretically analyze and prove the properties of energy conservation and convergence of numerical solution. (3) Based on the above two algorithms, propose an effective spectral method to solve quantum nonlinear dynamics equations, and use this algorithm to study the influence of nonlinear effect on quantum systems.