混合不连续伽辽金方法的多辛性
Multisymplecticity of hybridizable discontinuous Galerkin methods
中文总结 AI 辅助
本文研究了混合不连续伽辽金方法在求解 canonical Hamiltonian 系统时满足多辛守恒律的必要和充分条件，证明了多种常见有限元方法的混合版本满足这些条件，从而为结构保持离散化提供了方法支持。
中文摘要 AI 辅助
在本文中，我们证明了当应用于 canonical Hamiltonian 系统的偏微分方程时，混合不连续伽辽金（HDG）方法满足多辛守恒律的必要和充分条件。我们显示这些条件被多种最常用的有限元方法的“混合”版本满足，包括混合型、非顺应型和不连续伽辽金方法。（有趣的是，对于维度大于一的连续伽辽金方法，我们显示多辛性仅在较弱的意义上成立。）因此，这些通用的有限元方法可用于 canonical Hamiltonian 系统的 ODE 或 PDE 的结构保持离散化（或半离散化）。这为任意高阶方法在非结构化网格上建立多辛性奠定了基础。
英文摘要
In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin (HDG) method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the "hybridized" versions of several of the most commonly-used finite element methods, including mixed, nonconforming, and discontinuous Galerkin methods. (Interestingly, for the continuous Galerkin method in dimension greater than one, we show that multisymplecticity only holds in a weaker sense.) Consequently, these general-purpose finite element methods may be used for structure-preserving discretization (or semidiscretization) of canonical Hamiltonian systems of ODEs or PDEs. This establishes multisymplecticity for a large class of arbitrarily-high-order methods on unstructured meshes.
