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Integration of the EPDiff equation by particle methods∗∗∗∗∗∗

Published online by Cambridge University Press:  11 January 2012

Alina Chertock ,
Philip Du Toit  and
Jerrold Eldon Marsden
Alina Chertock
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, 27695 NC, USA. chertock@math.ncsu.edu
Philip Du Toit
Affiliation:
Control and Dynamical Systems, California Institute of Technology, Pasadena, 91125 CA, USA; pdutoit@cds.caltech.edu
Jerrold Eldon Marsden
Affiliation:
Control and Dynamical Systems, California Institute of Technology, Pasadena, 91125 CA, USA; jmarsden@caltech.edu

Abstract

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The purpose of this paper is to apply particle methods to the numerical solution of theEPDiff equation. The weak solutions of EPDiff are contact discontinuities that carrymomentum so that wavefront interactions represent collisions in which momentum isexchanged. This behavior allows for the description of many rich physical applications,but also introduces difficult numerical challenges. We present a particle method for theEPDiff equation that is well-suited for this class of solutions and for simulatingcollisions between wavefronts. Discretization by means of the particle method is shown topreserve the basic Hamiltonian, the weak and variational structure of the originalproblem, and to respect the conservation laws associated with symmetry under the Euclideangroup. Numerical results illustrate that the particle method has superior features in bothone and two dimensions, and can also be effectively implemented when the initial data ofinterest lies on a submanifold.

Keywords

Solitonspeakonsintegrable Hamiltonian systemsparticle methodsweak solutionsvariational principlemomentum mapsshallow water and internal waves

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