Discrete momentum maps for lattice EPDiff

Abstract

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.