多频高振荡哈密尔顿系统的三角渐近型保结构算法研究

Hamiltonian system is one of important system in dynamics because all earthy physical course without any loss of energy must be classified in Hamiltonian system. This research project concerns the research on trigonometrically asymptotic structure-preserving algorithms for multi-frequency highly oscillatory Hamiltonian systems with a principal frequency matrix. The solutions of these systems are remarkable multi-frequency highly oscillators and traditional numerical methods cannot produce satisfactory numerical results since they do not respect the special structure brought by the principal frequency matrix. Moreover, the analytic technique for traditional numerical methods cannot be applied to exploring the long-time behaviour of methods for highly oscillatory systems. Therefore, based on trigonometrical variation-of-constants formula and taking advantage of asymptotic methods, this research project firstly derives and formulates trigonometrically asymptotic Filon methods and trigonometrically asymptotic collocation methods. Then the error bounds, the long-time energy conservations and the resonance instabilities for the numerical methods are analyzed. All the trigonometrically asymptotic structure-preserving algorithms in this research project are constructed and analyzed based on trigonometrical variation-of-constants formula and make good use of the special structure brought by the principal frequency matrix. Therefore, the novel methods perform well in simulation of the exact solution and are more efficient and effective than traditional methods. Moreover, all the derived methods and their theoretical analysis avoid the decomposition, inversion and diagonalization of the principal frequency matrix. It is known that matrix decomposition, inversion and diagonalization usually not only bring additional computation but also result in new errors or other disadvantages especially when the dimension of the matrix is large. Therefore, the novel methods researched in this project are significant in long-time numerical computations.

哈密尔顿系统是动力系统的重要体系，一切真实的、耗散可忽略不计的物理过程都可表示成哈密尔顿系统。本项目以带有主频率矩阵的多频高振荡哈密尔顿系统为研究对象，研究高效三角渐近型保结构计算的理论与方法。因为此系统具有显著高振荡特性，而传统算法没有考虑到主频率矩阵所带来的特殊结构，所以传统算法不能有效地求解此系统，并且传统算法经典的分析方法不能用来分析高振荡系统长期积分后的行为性态。本项目以三角型常数变易公式为出发点，利用不同的渐近展开方法，构造三角渐近型Filon法和三角渐近型配置法；针对所得算法，分析研究算法的误差界、长期保能量性态和共振不稳定性。本项目所研究的算法，完全从三角型常数变易公式出发，充分利用主频率矩阵所带来的特殊结构，贴近精确解满足的格式，从而具有传统经典算法所不具备的高效性，并且算法的推导和分析直接依赖于主频率矩阵，避免矩阵分解、求逆和对角化，这对于长期数值计算具有重大意义。

哈密尔顿系统是动力系统中一类非常重要的体系，它广泛存在于应用科学的多个领域，例如应用数学、量子物理、经典力学、分子生物、化学、大气学、电子研究等等。在过去的二三十年，针对哈密顿系统的保结构算法得到众多科研工作者的广泛关注与研究。本项目针对带有主频率矩阵的多频高振荡哈密尔顿系统，研究了多种高效的保结构计算的理论与方法。多频高振荡哈密尔顿系统因为系统本身具有的高振荡性，所有传统的算法不能给出令人满意的结果。因此针对此系统，我们构造、分析和研究了多种高效的保结构算法及其重要性质。 .针对多频高振荡哈密尔顿系统，我们研究了新型三角傅里叶配置法并分析了算法的阶、收敛性、保能量性质和保二次不变量性质；构造了一类基于拉格朗日多项式的三角型配置法并研究了方法的性质；针对渐近展开方法，我们研究了算法系数函数的界并以此推导了算法的全局误差；研究了指数型傅里叶配置法及其保能量性质；构造了辛和对称算法来保持哈密顿系统的辛性和对称性；为了实际计算，我们分析讨论了方法的高效执行和实现。.项目执行中我们研究的所有保结构算法都是基于常数变易公式和多频高振荡系统中的频率矩阵进行构造与分析，因此正如论文中数值实验展示的结果一样，我们的新算法能很好的逼近精确解并且比传统算法更加高效。另外，在算法构造和分析中，我们避免了频率矩阵的分解、求逆和对角化，这对于长期计算具有重要的意义。该项目的进行为多频高振荡哈密顿系统提供了多种高效保结构算法以供广大科研工作者选择和应用，同时为其它类型微分方程数值算法的研究提供了新的思路与分析，对于高振荡哈密顿系统的数值研究具有重要的意义。