  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.
