格点EPDiff的离散动量映射
Discrete momentum maps for lattice EPDiff
AI总结：
本文通过变分粒子网格(VPM)方法对EPDiff方程进行空间离散，提出了约束变分原理并推导了离散动量映射。
摘要：
我们关注变分粒子网格(VPM)方法对原型流体方程EPDiff方程的空间离散化，EPDiff是微分同胚群（在$\mathbb{R}^d$或$d$维流形$\Omega$上）的Euler-Poincaré方程的简称。EPDiff方程允许测度值解，其动力学由微分同胚在$\mathbb{R}^d$嵌入子空间上的左、右作用的动量映射决定。本文研究了这些动力学的离散VPM模拟。我们的主要结果是：(i) VPM方法的变分公式，用拉格朗日粒子的约束变分原理表示，其速度被限制在分布$D_{\VPM}$上，该分布是$\Omega$上向量场李代数的有限维子空间；(ii) 固定欧拉网格上的相应约束变分原理，给出了Euler-Poincaré方程的离散版本；(iii) 微分同胚在解空间上的左、右作用的动量映射的离散版本。
英文摘要：
We focus on the spatial discretization produced by the Variational Particle-Mesh (VPM) method for a prototype fluid equation the known as the EPDiff equation}, which is short for Euler-Poincaré equation associated with the diffeomorphism group (of $\mathbb{R}^d$, or of a $d$-dimensional manifold $Ω$). The EPDiff equation admits measure valued solutions, whose dynamics are determined by the momentum maps for the left and right actions of the diffeomorphisms on embedded subspaces of $\mathbb{R}^d$. The discrete VPM analogs of those dynamics are studied here. Our main results are: (i) a variational formulation for the VPM method, expressed in terms of a constrained variational principle principle for the Lagrangian particles, whose velocities are restricted to a distribution $D_{\VPM}$ which is a finite-dimensional subspace of the Lie algebra of vector fields on $Ω$; (ii) a corresponding constrained variational principle on the fixed Eulerian grid which gives a discrete version of the Euler-Poincaré equation; and (iii) discrete versions of the momentum maps for the left and right actions of diffeomorphisms on the space of solutions.
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