高振荡量子动力系统的保结构算法与应用

This project focuses on systematic research on the theory, methods and applications of structure-preserving computation for highly oscillatory second-order quantum dynamical systems. Among the concerned stuctures of dynamical systems include invariants like Hamiltonian, symplecticity and multisympleticity, oscillation and high oscillation etc. The numerical integrators to be investigated are required to preserve the important physical properties of the exact solutions of the systems and they must have phase-lage order，dissipation order and algebraic order as high as possible and to have proper stability. Techinques of trigonometrical/exponential fitting and phase and amplification fitting are used to improve their ability of preserving structures; techniques of adaptive embedding and FSAL are used to upgrade their efficiency. Dirichlet's expansion is employed to approximate the solution of multivariate highly oscillatory problems. Modulated Fourier expansion is used to analyse the almost energy-preservation of numerical integrators. The AVF-type methods which avoid the computation of matrix exponential are used to preserve exactly the Hamiltonian. New multisymplectic integrators are developed for Hamiltonian PDEs. Spectrum and pseudospectrum methods are also considered. The effectiveness and efficiency of new methods are tested via numerical experiments. Applications involve the orbit problems, semi-discretized hyperbolic PDEs, molecular dynamical systems and quantum biological dynamical systems.

本项目系统研究高振荡二阶量子动力学系统的保结构计算的理论、方法及其应用。所关心的动力系统结构包括Hamilton量等不变量、辛与多辛性、振荡与高振荡性等，要求数值方法能够保持系统精确解的重要物理性质的同时，有尽可能高的相延迟阶、耗散阶和代数阶，并具有好的稳定性。三角拟合与指数拟合、相拟合与振幅拟合等技术提高方法的保结构性；变步长自适应嵌入技术与FSAL技术可提高方法的计算效率。用Dirichlet展开逼近多变量高振荡问题的解；modulated Fourier展开分析振荡系统数值积分方法的几乎保能量性；用避免矩阵指数计算的AVF型方法以精确保持原问题的Hamilton量；发展求解Hamiltonian PDE的新的多辛方法；探索针对量子力学方程的谱和伪谱方法。通过数值实验检验新方法的有效性与高效性；应用问题涉及轨道问题、半离散双曲型PDEs，分子动力系统、量子生物动力系统。

在现代数值分析的前沿领域中高振荡微分方程保结构算法的研究是一个非常困难的问题。本项目系统研究高振荡二阶量子动力学系统的保结构计算的理论、方法及其应用。所关心的动力系统结构包括Hamilton量等不变量、辛与多辛性、振荡与高振荡性等，要求数值方法能够保持系统精确解的重要物理性质的同时，有尽可能高的相延迟阶、耗散阶和代数阶，并具有好的稳定性。三角拟合与指数拟合、相拟合与振幅拟合等技术提高方法的保结构性；变步长自适应嵌入技术与FSAL技术可提高方法的计算效率。用Dirichlet展开逼近多变量高振荡问题的解；modulated Fourier展开分析振荡系统数值积分方法的几乎保能量性；用避免矩阵指数计算的AVF型方法以精确保持原问题的Hamilton量；发展求解Hamiltonian PDE的新的多辛方法；探索针对量子力学方程的谱和伪谱方法。通过数值实验检验新方法的有效性与高效性；应用问题涉及轨道问题、半离散双曲型PDEs，分子动力系统、非线性哈密顿系统。