The purpose of this paper is to apply particle methods to the numerical solution of the
EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry
momentum so that wavefront interactions represent collisions in which momentum is
exchanged. This behavior allows for the description of many rich physical applications,
but also introduces difficult numerical challenges. We present a particle method for the
EPDiff equation that is well-suited for this class of solutions and for simulating
collisions between wavefronts. Discretization by means of the particle method is shown to
preserve the basic Hamiltonian, the weak and variational structure of the original
problem, and to respect the conservation laws associated with symmetry under the Euclidean
group. Numerical results illustrate that the particle method has superior features in both
one and two dimensions, and can also be effectively implemented when the initial data of
interest lies on a submanifold.