【作者】 颜娜；

【导师】 周钢；

【作者基本信息】 上海交通大学 ， 计算数学， 2010， 硕士

【摘要】 自从冯康和Ruth提出了求解Hamilton系统的辛算法后，国内外学者对辛和多辛算法的研究已经取得了很大的成就。辛几何算法的主要特点有：①保持原有系统的结构，具有长期的跟踪能力，因此在长期计算中可以验证计算结果的准确性；②挖掘辛几何规律，若能赋予相应的物理意义，便有可能发现和发掘新的自然规律。保守体系应该保辛，Hamilton体系是保守体系，因此有必要探讨应用辛算法来解决各类Hamilton系统中的规律挖掘问题。本文2～4章是学术研究论文，作者在文中取得了一系列的创新成果，主要是从理论上分析了一般的字母型偶数阶线性Hamilton系统的辛算法飘移性及其规律。利用辛算法的保结构性，将迭代矩阵作同构变形，通过比对精确解与近似解，得到飘移量公式。建立了纠飘算法，相应的数值计算结果令人满意。本文的第5章，是作者按照“上海交通大学数学系硕士研究生毕业要求”的条例完成的，是在阅读、理解大量科技文献后经思考、提炼而撰写的综合报告，主要综述了辛算法在小参数摄动、非线性ODE摄动、振动系统、最优控制系统中的应用背景及其辛精细与辛RK算法。主要工作是将不同背景下的实际问题转化为线性Hamilton系统，选取合适的辛差分格式求解。数值算例表明了:辛算法在线性Hamilton系统中的相位飘移性质是具有普适性的。更多还原

【Abstract】 Scholars at home and abroad have scored great successes insymplectic and multi-symplectic methods since Fengkang and Ruthproposed symplectic algorithm for Hamilton systems.One of the most important characteristics of the algorithm ofsymplectic geometry is its structure conservation and long-term trackingability. So its accuracy of the results can be shown in secular computing.The other is its symplectic geometry laws. We may discover newnature laws with the given physical meaning.A coservative system should be symplectic conservative. Hamiltonsystem is a conservative system so symplectic algorithm can be used inthe solutions to the research on mining symplectic laws of Hamiltonsystems.This paper 2～4 chapters were the dissertation research the author hasyielded a series of innovation results in this paper. The theoretical proofof phase shift is proposed aiming at the general type of letters for evevorder linear Hamilton systems with periodic solutions mainly. The phaseshift formulas are given by matching exact solutions and approximatesolutions and using homogeneous deformation for interative matrixaccording to the structive preserving algorithms of symplectic geometry.The essence of phase shift of symplectic algotithm and the correctionshift method are given. Finally a numerical result demonstrate phase shifeof symplectic algorithm. The fifth chapter mainly are defer to“Shanghaijiaotong University math department master the graduatestudent to raise the rule”to complete. After reading and understandingmassive science and technology literature the general report which after the ponder the refinement completes. The fifth chapter which is thegeneral report emphasizes that the application backgrounds of symplecticalgorithm should be considered in small parameter perturbation method,the perturbation of nonlinear systems in ODE, vibration systems andoptimal control systems besides precise symplectic algorithm andsymplectic Runge-Kutta methods. The main idea is to transform thedifferent problems into linear Hamilton systems and solve them withsuitable symplectic difference schemes. Numerical analyses are carriedout to illustrate phase shift of symplectic algorithm in the solution oflinear Hamilton systems.更多还原