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.
Mots-clés : 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 },
pages = {515--534},
publisher = {EDP-Sciences},
volume = {46},
number = {3},
year = {2012},
doi = {10.1051/m2an/2011054},
mrnumber = {2877363},
zbl = {1272.65079},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011054/}
}
TY - JOUR AU - Chertock, Alina AU - Toit, Philip Du AU - Marsden, Jerrold Eldon TI - Integration of the EPDiff equation by particle methods JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 515 EP - 534 VL - 46 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011054/ DO - 10.1051/m2an/2011054 LA - en ID - M2AN_2012__46_3_515_0 ER -
%0 Journal Article %A Chertock, Alina %A Toit, Philip Du %A Marsden, Jerrold Eldon %T Integration of the EPDiff equation by particle methods %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 515-534 %V 46 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011054/ %R 10.1051/m2an/2011054 %G en %F 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 , Special volume in honor of Professor David Gottlieb. Numéro spécial, Tome 46 (2012) no. 3, pp. 515-534. doi : 10.1051/m2an/2011054. http://www.numdam.org/articles/10.1051/m2an/2011054/
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