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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
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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.

Keywords

Solitonspeakonsintegrable Hamiltonian systemsparticle methodsweak solutionsvariational principlemomentum mapsshallow water and internal waves
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 3: Special volume in honor of Professor David Gottlieb , May 2012 , pp. 515 - 534
Copyright
© EDP Sciences, SMAI, 2012

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