Integration of the EPDiff equation by particle methods^{∗,∗∗,∗∗∗}

¹ Department of Mathematics, North Carolina State University, Raleigh, 27695 NC, USA
chertock@math.ncsu.edu
² Control and Dynamical Systems, California Institute of Technology, Pasadena, 91125 CA, USA
pdutoit@cds.caltech.edu
³ Control and Dynamical Systems, California Institute of Technology, Pasadena, 91125 CA, USA
jmarsden@caltech.edu

Received: 23 September 2009

Abstract

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.