Discrete Variational Derivative Methods for the EPDiff equation

TL;DR

This paper develops structure-preserving numerical schemes for the 2D EPDiff equation using Discrete Variational Derivative Methods, demonstrating their effectiveness in conserving physical quantities and handling wave interactions.

Contribution

It introduces three novel DVDM-based schemes for the EPDiff equation, emphasizing structure preservation and applicability to two-dimensional problems.

Findings

Schemes effectively preserve energy and momentum.

Numerical experiments show good convergence.

Successful simulation of wave front interactions.

Abstract

The aim of this paper is the derivation of structure preserving schemes for the solution of the EPDiff equation, with particular emphasis on the two dimensional case. We develop three different schemes based on the Discrete Variational Derivative Method (DVDM) on a rectangular domain discretized with a regular, structured, orthogonal grid. We present numerical experiments to support our claims: we investigate the preservation of energy and linear momenta, the reversibility, and the empirical convergence of the schemes. The quality of our schemes is finally tested by simulating the interaction of singular wave fronts.