【作者】 罗顺；

【导师】 周钢；

【作者基本信息】 上海交通大学 ， 计算数学， 2007， 硕士

【摘要】 自钟万勰院士1994年提出齐次线性自治动力系统的精细算法HPD以来,这一计算力学、工程应用与计算数学的学科交叉点迅速发展,已成为学术热点。本文基于已有的成果,与多辛算法相结合,将精细算法引向深入,开展了MKS PDE的MSHPD算法的系列性研究,并讨论了精细算法在时变齐次线性系统中的应用,得到了令人满意的结果。本文的创新工作主要有以下几个方面(1)就MKS PDE讨论了多辛算法、精细算法及其复合算法,提出MSHPD算法。多辛算法的优点在于能保留物理规律,精细算法的优点则在于其能保留高精度的长效性,本文给出了求解MKS PDE的MSHPD算法的一般步骤,该算法兼具保留物理规律和高精度长效性的优点。算例表明:MSHPD算法的计算结果令人满意。(2)将MSHPD算法应用于实际问题,集中讨论了MKdV方程和梁振动方程。在既保持物理规律又保持高精度长效性方面,数值结果令人满意。(3)进一步将MSHPD算法应用于复系数时变系统,给出了复系数时变系统的MSHPD算法,数值结果令人满意。(4)就两类特殊时变线性系统提出了HHPD-P齐次扩容精细算法,数值结果令人满意。(5)就MSHPD算法中特殊传递矩阵,设计出一种初参数求逆算法,利用该算法可实现快速求逆。(见附录)(6)建立计算方法与图模型的对应关系,尝试图模型在系统设计中的应用。以Duffing方程及其控制为例探讨如何通过建立图模型,由算法设计出相容的“数学模型”与更新计算方法。(见附录)(7)建立常见的计算方法的图模型,通过图模型可直接判断算法的兼容稳更多还原

【Abstract】 The High Precision Direct integration method has been developing quickly and becoming an academic focus since Mr. Zhong Wanxie proposed it in 1994.It intersects of computational mechanics, engineering application and computational mathematics. Based on achieved results, this paper focuses on the combination of multi-symplectic algorithm and high precision algorithm, tries to put high precision multi-symplectic algorithm forward. The paper carries on a series of researches about the new algorithm named MSHPD algorithm as well as some discussion about its application in variable coefficients systems.The paper’s mainly innovations as following(1)Discuss multi-symplectic algorithm, high precision algorithm and its combined algorithm on MKS PDE and bring forward a new algorithm named MSHPD algorithm. We know, multi-symplectic algorithm shows the advantage at remaining physical law while high precision algorithm shows its at long term effect of high precision. The paper describes the general steps of MSHPD algorithm to MKS PDE. The high precision multi-symplectic algorithm mentioned in paper has the both advantages. The numerical examples show the algorithm is satisfied.更多还原