Discrete momentum maps for lattice EPDiff

Details Discrete momentum maps for lattice EPDiff Discrete momentum maps for lattice EPDiff

2006-02-14

arxiv.org/abs/math/0602296v1

Mathematics

Numerical Analysis

Scientific paper

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-Poincar\'e equation associated with the diffeomorphism group (of $\mathbb{R}^d$, or of a $d$-dimensional manifold $\Omega$). 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 $\Omega$; (ii) a corresponding constrained variational principle on the fixed Eulerian grid which gives a discrete version of the Euler-Poincar\'e equation; and (iii) discrete versions of the momentum maps for the left and right actions of diffeomorphisms on the space of solutions.

Affiliated with

Also associated with

No associations

Rating

Discrete momentum maps for lattice EPDiff does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Discrete momentum maps for lattice EPDiff, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Discrete momentum maps for lattice EPDiff will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-568334