Geometric Discretization of the EPDiff Equations

TL;DR

This paper introduces a new geometric discretization method for infinite-dimensional systems, specifically applied to the EPDiff equation, extending previous methods and demonstrating its effectiveness through numerical results.

Contribution

It generalizes existing discretization techniques to all vector fields and employs a pseudospectral approach, broadening the applicability to the EPDiff equation.

Findings

Successful application to 1D EPDiff equation

Numerical results demonstrate method effectiveness

Extension of discretization to all vector fields

Abstract

The main objective of this paper is to develop a general method of geometric discretization for infinite-dimensional systems and apply this method to the EPDiff equation. The method described below extends one developed by Pavlov et al. for incompressible Euler fluids. Here this method is presented in a general case applicable to all, not only divergence-free, vector fields. Also, a different (pseudospectral) representation of the velocity field is used. We will apply this method to the one-dimensional EPDiff equation and present numerical results.