含耗散项偏微分方程的多辛Birkhoff结构
Multi-symplectic Birkhoffian Structure for PDEs with Dissipation Terms
AI总结：
本文提出将哈密顿系统的多辛形式推广至含耗散项的偏微分方程，通过引入显式依赖时间的辛结构定义多辛Birkhoff方程及耗散律，并探讨了多辛积分器概念。
摘要：
哈密顿偏微分方程的多辛形式为几何数值格式提供了一个通用框架，该框架可保持辛性守恒的离散版本。对于含耗散项的系统或偏微分方程，这一框架此前从未被扩展。本文提出一种新的扩展，将哈密顿系统的多辛形式推广到不具有显著能量和动量守恒性质的耗散系统。核心思想是：所考虑的偏微分方程为一阶类型，其辛结构显式依赖于时间变量，并可分解为分别代表空间和时间方向的不同分量。这自然引出了多辛Birkhoff方程的定义，将其作为一种多辛结构，并由此构造出多辛耗散律。我们证明，这一定义有助于更深入地理解泛函原理与偏微分方程之间的关系。此外，还讨论了多辛积分器的概念。
英文摘要：
The multi-symplectic form for Hamiltonian PDEs leads to a general framework for geometric numerical schemes that preserve a discrete version of the conservation of symplecticity. The cases for systems or PDEs with dissipation terms has never been extended. In this paper, we suggest a new extension for generalizing the multi-symplectic form for Hamiltonian systems to systems with dissipation which never have remarkable energy and momentum conservation properties. The central idea is that the PDEs is of a first-order type that has a symplectic structure depended explicitly on time variable, and decomposed into distinct components representing space and time directions. This suggest a natural definition of multi-symplectic Birkhoff's equation as a multi-symplectic structure from that a multi-symplectic dissipation law is constructed. We show that this definition leads to deeper understanding relationship between functional principle and PDEs. The concept of multi-symplectic integrator is also discussed.
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