振荡哈密尔顿波方程的几何数值积分

This project focuses on geometric numerical integration (i.e., structure- preserving algorithm) of oscillatory Hamiltonian wave equations. Such systems arise in various fields such as physics, astronomy, material etc, which are of great importance in theory and applications as well. Among the concerned structures of the system are oscillation, symplecticity and multi-sympleticity, energy conservation etc. The numerical integrators to be investigated are required to preserve the above properties of the system as much as possible and stability of the numerical methods will be considered as well. The key point is that the integrators constructed are adapted to the oscillatory structure while preserving the properties of the system such as symplecticity and energy. The tasks of the project are as follows:1) Construct proper spatial semi-discretization towards different structures of wave equations, which results in a system of Hamiltonian ODEs, then, for the resulted system, AAVF formula and extended discrete gradient method both of which are adapted to the oscillation can be used to obtain energy-preserving schemes. The spatial semi-discretization of the conservative wave equations are considered from two different points of view: a) Consider directly the wave equation, discretize directly the spatial derivative by using spectral discretization with trigonometric (Fourier, Lagrangian) basis or finite element method. b) Discretize the energy function of integral form using a consistent approximation by approaches such as discrete Fourier transform and discrete variational derivative etc. 2) Take account of oscillatory feature of the system, new multi-symplectic integrators for oscillatory Hamiltonian wave equations will be developed for higher dimensional case. 3) Based on the operator-variation-of-constants formula, we will investigate the semi-continuous schemes for oscillatory wave equations. The symplecticity and multi-symplecticity of the obtained schemes will be explored.

本项目研究振荡哈密尔顿波方程几何数值积分(即保结构计算)。这类方程广泛存在于物理、天文、材料等领域，具有重要的理论和应用价值。所关心的系统结构包括振荡性、辛与多辛性、能量守恒性等。数值方法在保持系统辛性或能量的同时，更关键的是适应振荡结构，并具有好的稳定性。本项目包含如下内容：1) 针对波方程的结构特点构造恰当的空间半离散，对半离散得到的哈密尔顿常微分系统，利用适应于振荡结构的AAVF方法、扩展的离散梯度法分析构造守恒格式；空间半离散从两个角度考虑:a) 从方程本身出发，对空间导数利用三角( Fourier， Lagrange) 基的谱离散方法或有限元逼近。b) 从系统能量出发，采用离散Fourier变换、离散变分导数等方法离散积分形式的能量函数；2）结合系统的振荡性，发展求解高维哈密尔顿波方程的多辛方法；3）根据算子常数变易公式，构建求解振荡波方程半连续型数值格式并讨论格式的辛和多辛性。

本研究针对守恒/耗散型振荡哈密尔顿系统分别研究了保持系统振荡特性及保持系统能量守恒/耗散性质的算法。对于能量守恒系统，空间方向分别采用有限差分、有限元手段进行离散，时间方向利用能量守恒算法，从而很好地近似原系统能量，并长时间保持；对于能量耗散系统，空间方向分别利用有限差分和谱离散手段对能量函数进行离散逼近，时间方向用修正的扩展离散梯度方法构造了同时保持系统振荡特性和能量耗散性质的数值格式，并使得数值能量与精确能量耗散速度一致。 针对变系数的齐次/非齐次线性波动方程，结合文献中提出的算子常数变易公式，构造了不依赖于方程维数的紧凑格式，并讨论了一类特殊变系数波动方程解的存在唯一性。针对系统右端函数含有阻尼项的二阶振荡常微分方程，研究得到了一个新的测试方程，给出了新的色散耗散定义，在处理振荡微分方程时，能对格式的有效性有更深刻的理解。本研究同时证明了在给定网格下，微分方程中的空间一阶导数项任意高阶反对称微分矩阵的存在性，此高阶反对称微分矩阵离散对保持数值格式的稳定性具有重要的意义。