Discrete Embeddings for Lagrangian and Hamiltonian Systems
Discrete Embeddings for Lagrangian and Hamiltonian Systems
复制标题
拉格朗日和哈密顿系统的离散嵌入
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
C. Pierre
中科院分区：
文献类型：
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作者：
J. Cresson;I. Greff;C. Pierre
本文的主题是研究离散化问题的变分性质守恒性。准确地说，我们对拉格朗日或哈密顿结构感兴趣，因此对与最小作用量原理有关的变分问题感兴趣。考虑由变分原理导出的偏微分方程(PDE)。一个自然的问题是，当离散PDE时，是否在离散水平上保留了这种结构。为了解决这个问题，引入了连贯的概念。微分方程(转换最小作用量原理的偏微分方程组)和变分结构都可以嵌入到离散层。这为原始问题提供了两个离散嵌入。如果这些程序最终提供了相同的离散问题，我们将说离散化是连贯的。我们的目的用泊松问题来说明。研究了各种经典离散化下拉格朗日结构离散嵌入的相干性。对于哈密顿结构，我们证明了离散哈密顿量与泊松问题混合形式的离散化之间的一致性。
The topic of this paper is to study the conservation of variational properties for a given problem when discretising it. Precisely, we are interested in Lagrangian or Hamiltonian structures and thus with variational problems attached to a least action principle. Consider a partial differential equation (PDE) deriving from a variational principle. A natural question is to know whether this structure is preserved at the discrete level when discretising the PDE. To address this question, a concept of coherence is introduced. Both the differential equation (the PDE translating the least action principle) and the variational structure can be embedded at the discrete level. This provides two discrete embeddings for the original problem. If these procedures finally provide the same discrete problem, we will say that the discretisation is coherent. Our purpose is illustrated with the Poisson problem. Coherence for discrete embeddings of Lagrangian structures is studied for various classical discretisations. For Hamiltonian structures, we show the coherence between a discrete Hamiltonian and the discretisation of the mixed formulation of the Poisson problem.