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离散辛算法及其要物理中的应用

批准号10171096 学科分类土壤化学 ( D070103 )
项目负责人周东美 负责人职称 依托单位中国科学院理论物理研究所
资助金额0.00
万元
项目类别面上项目 研究期限2002 年 01 月 01 日 至
2004 年 12 月 01 日
中文主题词欧拉-拉格朗日上同调;推广的刘维定理;差分变分原理;保能量辛算法;辛有限元格式
英文主题词Euler-Lagrange cohomology; Generalized Liouville's theorem; difference discrete variation; energy-preserving symplectic algorithm; symplectic FEM.

摘要

中文摘要 研究(相)空间离散的辛几何与辛算法以及保结构算法、多辛算法及其在物理中的应用。探讨空间差分格式、有限元方法的离散几何结构,尝试建立它们的辛算法或保结构算法;考察其在物理中的应用。由于大量的哈密顿系统是空间离散的,因此如果能够探讨这些系统的离散(辛)几何并深度建立其保结构算法,无疑是具有重要意义的。..
英文摘要 We've found that on symplectic manifolds there exists a series of Euler-Lagrange cohomology groups different from both de Rham and harmonic cohomology in general and proposed a set of general volume-preserving equations that contain the Hamiltonian equations as a special case. Further we've generalized the famous Liouville's theorem in classical mechanics and linked the Noether theorem with cohomology..For the difference discrete systems, we've proposed the difference discrete variational principle with fixed or variable step-lengths, in which the both discrete dependent variables and their differences are taken as independent variables in variation, and introduced the discrete Legendre transformation as well as the discrete version of the E-Lcohomology. We've applied these principle and methods to the symplectic/ multisymplectic algorithms so as to the discrete Hamiltonian and Lagrangian formalism can be transformed from one the other. We've also found the necessary and sufficient condition for the symplectic/multisymplectic preserving, which is not on the solution space of the system in general. Further, we've taken the difference discrete variational principle with variable step-lengths to get the (discrete) energy-preserving symplectic/multisymplectic algorithms..We've also explored the relation between the structure-preserving algorithm and the finite element method/mixed finite element method. By means of the difference discrete variational principle, discrete Legendre transformation and discrete version of the E-L cohomology, for a kind of regular meshes we've found there is a kind of symplectic/multisymplectic structures with corresponding symplectic/multisymplectic finite element method/mixed finite element method schemes.?We've also proposed a new discrete symplectic algorithm inspired by the finit element method.
结题摘要 我们发现在作为经典力学中的相空间的辛流形上有可能存在不同于de Rham上同调和调和形式的上同调,称之为Euler-Lagrange(EL)上同调;提出相空间上保体积的一般方程,通常的正则方程是其特殊形式;进而推广了经典力学中著名的刘维定理;同时把Noether定理与上同调联系起来。 对于差分离散力学和场论系统,将离散变量及其差分都作为离散变分变量,提出等步长或变步长的差分离散变分原理。引入离散Legendre变换将离散的哈氏形式与拉氏形式统一起来。引入EL上同调的差分离散形式。并将这些原理和方法应用于辛算法和多辛算法,统一了辛和多辛算法的哈氏形式和拉氏变分形式,得到保辛/多辛结构的充要条件,并非通常认为的在辛/多辛格式的解空间上。进而,将辛和多辛算法推广到变步长,得到保能量的辛和多辛算法。 利用这些原理和方法探讨了保结构算法与有限元和混合有限元方法之间的联系。利用差分离散变分原理、离散Legendre变换和EL上同调的离散形式,对于正规网格找到一类有限元和混合有限元的辛或多辛结构,以及相应的辛有限元/混合有限元格式。反之,利用有限元离散相空间,得到一类新的算法,称为有限元-辛算法。

成果

序号 标题 类型 作者
1 Difference discrete variational principle in discrete mechanics and symplectic algorithm, 期刊 X.D.Luo|H.-Y. Guo|Y.Q. Li|K. Wu|
2 General volume-preserving mechanical systems, 期刊 B. Zhou|H.-Y. Guo|K. Wu|
3 On general volume-preserving mechanical systems via cohomology 期刊 B. Zhou|H.-Y. Guo|K. Wu|
4 Difference discrete variational principle, Euler-Lagrange cohomology and symplectic, multisymplectic structures II: Euler-Lagrange Cohomology 期刊 H.-Y. Guo|Y.Q. Li|K. Wu|S.K. Wang|
5 The Euler-Lagrange cohomology and general volume-preserving systems, 期刊 H.-Y. Guo|J.Z. Pan|K. Wu|B. Zhou|

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