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Integration of the EPDiff equation by particle methods

Chertock, Alina and Du Toit, Philip and Marsden, Jerrold Eldon (2012) Integration of the EPDiff equation by particle methods. ESAIM-Mathematical Modelling and Numerical Analysis, 46 (3). pp. 515-534. ISSN 0764-583X. http://resolver.caltech.edu/CaltechAUTHORS:20120326-111126626

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1051/m2an/2011054 DOIUNSPECIFIED
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=8476145PublisherUNSPECIFIED
Additional Information:© 2012 EDP Sciences, SMAI. Received September 23, 2009. Published online January 11, 2012. Published online by Cambridge University Press: 11 January 2012. The research of A. Chertock is partially supported by NSF grant DMS-0712898. The research of P. Du Toit is partially supported by AFOSR contract FA9550-08-1-0173. The research of J. E. Marsden is partially supported by AFOSR contract FA9550-08-1-0173.
Funders:
Funding AgencyGrant Number
NSFDMS-0712898
Air Force Office of Scientific Research (AFOSR)FA 9550-08-1-0173
Subject Keywords:Solitons, peakons, integrable Hamiltonian systems, particle methods, weak solutions, variational principle, momentum maps, shallow water and internal waves.
Classification Code:MSC: 35708, 37K10, 65M25, 74J35, 76B15.
Record Number:CaltechAUTHORS:20120326-111126626
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20120326-111126626
Official Citation:Integration of the EPDiff equation by particle methods Alina Chertock, Philip Du Toit and Jerrold Eldon Marsden ESAIM: Mathematical Modelling and Numerical Analysis / Volume 46 / Issue 03, May 2012, pp 515 - 534 Published online by Cambridge University Press: 11 January 2012 DOI:10.1051/m2an/2011054
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:29844
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:26 Mar 2012 18:38
Last Modified:26 Dec 2012 14:59

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