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

Alina Chertock; Philip Du Toit; Jerrold Eldon Marsden

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 3, page 515-534
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topChertock, Alina, Toit, Philip Du, and Marsden, Jerrold Eldon. "Integration of the EPDiff equation by particle methods∗∗∗∗∗∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.3 (2012): 515-534. <http://eudml.org/doc/222177>.

@article{Chertock2012,

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.},

author = {Chertock, Alina, Toit, Philip Du, Marsden, Jerrold Eldon},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Solitons; peakons; integrable Hamiltonian systems; particle methods; weak solutions; variational principle; momentum maps; shallow water and internal waves; solitons; Euler-Poincaré differential equation; numerical results},

language = {eng},

month = {1},

number = {3},

pages = {515-534},

publisher = {EDP Sciences},

title = {Integration of the EPDiff equation by particle methods∗∗∗∗∗∗},

url = {http://eudml.org/doc/222177},

volume = {46},

year = {2012},

}

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

DA - 2012/1//

PB - EDP Sciences

VL - 46

IS - 3

SP - 515

EP - 534

AB - 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.

LA - eng

KW - Solitons; peakons; integrable Hamiltonian systems; particle methods; weak solutions; variational principle; momentum maps; shallow water and internal waves; solitons; Euler-Poincaré differential equation; numerical results

UR - http://eudml.org/doc/222177

ER -

## References

top- M.S. Alber, R. Camassa, Y.N. Fedorov, D.D. Holm and J.E. Marsden, The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE’s of shallow water and Dym type. Commun. Math. Phys.221 (2001) 197–227.
- R. Artebrant and H.J. Schroll, Numerical simulation of Camassa-Holm peakons by adaptive upwinding. Appl. Numer. Math.56 (2006) 695–711.
- R. Beals, D.H. Sattinger and J. Szmigielski, Peakon-antipeakon interaction. J. Nonlin. Math. Phys.8 (2001) 23–27; Nonlinear evolution equations and dynamical systems, Kolimbary (1999).
- R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett.71 (1993) 1661–1664.
- R. Camassa, D.D. Holm and J.M. Hyman, A new integrable shallow water equation. Adv. Appl. Mech.31 (1994) 1–33.
- R. Camassa, J. Huang and L. Lee, On a completely integrable numerical scheme for a nonlinear shallow-water wave equation. J. Nonlin. Math. Phys.12 (2005) 146–162.
- R. Camassa, J. Huang and L. Lee, Integral and integrable algorithms for a nonlinear shallow-water wave equation. J. Comput. Phys.216 (2006) 547–572.
- S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett.81 (1998) 5338–5341.
- A. Chertock and A. Kurganov, On a practical implementation of particle methods. Appl. Numer. Math.56 (2006) 1418–1431.
- A. Chertock and D. Levy, Particle methods for dispersive equations. J. Comput. Phys.171 (2001) 708–730.
- A. Chertock and D. Levy, A particle method for the KdV equation. J. Sci. Comput.17 (2002) 491–499.
- A.J. Chorin, Numerical study of slightly viscous flow. J. Fluid Mech.57 (1973) 785–796.
- G.M. Coclite, K.H. Karlsen and N.H. Risebro, A convergent finite difference scheme for the Camassa-Holm equation with general H1 initial data. SIAM J. Numer. Anal.46 (2008) 1554–1579.
- A. Cohen and B. Perthame, Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal.32 (2000) 616–636 (electronic).
- G.-H. Cottet and P.D. Koumoutsakos, Vortex methods. Cambridge University Press, Cambridge (2000).
- G.-H. Cottet and S. Mas-Gallic, A particle method to solve transport-diffusion equations, Part 1 : the linear case. Tech. Report 115, Ecole Polytechnique, Palaiseau, France (1983).
- G.-H. Cottet and S. Mas-Gallic, A particle method to solve the Navier-Stokes system. Numer. Math.57 (1990) 805–827.
- P. Degond and S. Mas-Gallic, The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity. Math. Comput.53 (1989) 485–507.
- P. Degond and S. Mas-Gallic, The weighted particle method for convection-diffusion equations. II. The anisotropic case. Math. Comput.53 (1989) 509–525.
- P. Degond and F.-J. Mustieles, A deterministic approximation of diffusion equations using particles. SIAM J. Sci. Statist. Comput.11 (1990) 293–310.
- S. Gottlieb, C.-W. Shu and E. Tadmor, High order time discretization methods with the strong stability property. SIAM Rev.43 (2001) 89–112.
- O.-H. Hald, Convergence of vortex methods, Vortex methods and vortex motion. SIAM, Philadelphia, PA (1991) 33–58.
- A.N. Hirani, J.E. Marsden and J. Arvo, Averaged Template Matching Equations, EMMCVPR, Lecture Notes in Computer Science2134. Springer (2001) 528–543.
- H. Holden and X. Raynaud, Convergence of a finite difference scheme for the Camassa-Holm equation. SIAM J. Numer. Anal.44 (2006) 1655–1680 (electronic).
- H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-Holm equation based on multipeakons. Discrete Contin. Dyn. Syst.14 (2006) 505–523.
- D.D. Holm and J.E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation, The breadth of symplectic and Poisson geometry, Progr. Math.232. Birkhäuser Boston, Boston, MA (2005) 203–235.
- D.D. Holm and M.F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst.2 (2003) 323–380 (electronic).
- D.D. Holm and M.F. Staley, Interaction dynamics of singular wave fronts, under “Recent Papers” at . URIhttp://cnls.lanl.gov/~staley/
- D.D Holm, J.T. Ratnanather, A. Trouvé and L. Younes, Soliton dynamics in computational anatomy. NeuroImage23 (2004) S170–S178.
- H.-P. Kruse, J. Scheurle and W. Du, A two-dimensional version of the Camassa-Holm equation, Symmetry and perturbation theory. World Sci. Publ., Cala Gonone, River Edge, NJ (2001) 120–127.
- A.K. Liu, Y.S. Chang, M.-K. Hsu and N.K. Liang, Evolution of nonlinear internal waves in the east and south China Sea. J. Geophys. Res.103 (1998) 7995–8008.
- J.E. Marsden and T.S. Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics17, 2nd edition. Springer-Verlag, New York (1999).
- R.I. McLachlan and P. Atela, The accuracy of symplectic integrators. Nonlinearity5 (1992) 541–562.
- R. McLachlan and S. Marsland, N-particle dynamics of the Euler equations for planar diffeomorfism. Dyn. Syst.22 (2007) 269–290.
- P.-A. Raviart, An analysis of particle methods. Numerical methods in fluid dynamics (Como, 1983), Lecture Notes in Math.1127. Springer, Berlin (1985) 243–324.
- G.D. Rocca, M.C. Lombardo, M. Sammartino and V. Sciacca, Singularity tracking for Camassa-Holm and Prandtl’s equations. Appl. Numer. Math.56 (2006) 1108–1122.
- S.F. Singer, Symmetry in mechanics : A gentle, modern introduction. Birkhäuser Boston Inc., Boston, MA (2001).
- Y. Xu and C.-W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation. SIAM J. Numer. Anal.46 (2008) 1998–2021.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.