Particle methods for the EPDiff equation

Dr. Alina Chertock, North Carolina State University

We consider a model of active fluid transport described by an  evolutionary equation, known as the the Euler-Poincaré (EPDiff) equation.  The EPDiff equation arises in many scientific applications. In particular, it appears in the nonlinear dynamic of shallow water waves, and coincides, for example, with the Camassa-Holm equation of shallow water in 1-D and 2-D, and with the averaged template matching equation for computer vision in higher dimensions. The EPDiff singular + solutions are  contact discontinuities, called peakons. The key feature of the peakons is that they carry momentum; so the wave front interactions they represent are collisions, in which momentum is exchanged.  This is very reminiscent to the KdV solitons behavior in 1-D.

We numerically investigate the EPDiff dynamics of contact interactions using particle methods. We show that he discretization by means of the particle method preserves the basic Hamiltonian, the weak and variational structure of the original problem and respects the conservation laws associated to the symmetry under the Euclidean group. Both one- and two-dimensional numerical examples will be presented.