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.

Classification: 35708, 37K10, 65M25, 74J35, 76B15

Keywords: solitons, peakons, integrable hamiltonian systems, particle methods, weak solutions, variational principle, momentum maps, shallow water and internal waves

@article{M2AN_2012__46_3_515_0, author = {Chertock, Alina and Toit, Philip Du and Marsden, Jerrold Eldon}, title = {Integration of the EPDiff equation by particle methods}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {46}, number = {3}, year = {2012}, pages = {515-534}, doi = {10.1051/m2an/2011054}, zbl = {1272.65079}, mrnumber = {2877363}, language = {en}, url = {http://www.numdam.org/item/M2AN_2012__46_3_515_0} }

Chertock, Alina; Toit, Philip Du; Marsden, Jerrold Eldon. Integration of the EPDiff equation by particle methods. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, pp. 515-534. doi : 10.1051/m2an/2011054. http://www.numdam.org/item/M2AN_2012__46_3_515_0/

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