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.
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {46},
year = {2012},
pages = {515-534},
doi = {10.1051/m2an/2011054},
mrnumber = {2877363},
zbl = {1272.65079},
language = {en},
url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 515-534. doi : 10.1051/m2an/2011054. http://gdmltest.u-ga.fr/item/M2AN_2012__46_3_515_0/
[1] , , , and , 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. | MR 1846907 | Zbl 1001.37062
[2] and , Numerical simulation of Camassa-Holm peakons by adaptive upwinding. Appl. Numer. Math. 56 (2006) 695-711. | MR 2211502 | Zbl 1156.65313
[3] , and , Peakon-antipeakon interaction. J. Nonlin. Math. Phys. 8 (2001) 23-27; Nonlinear evolution equations and dynamical systems, Kolimbary (1999). | MR 1821503 | Zbl 0977.35106
[4] and , An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993) 1661-1664. | MR 1234453 | Zbl 0972.35521
[5] , and , A new integrable shallow water equation. Adv. Appl. Mech. 31 (1994) 1-33. | Zbl 0808.76011
[6] , and , On a completely integrable numerical scheme for a nonlinear shallow-water wave equation. J. Nonlin. Math. Phys. 12 (2005) 146-162. | MR 2117177
[7] , and , Integral and integrable algorithms for a nonlinear shallow-water wave equation. J. Comput. Phys. 216 (2006) 547-572. | MR 2235383 | Zbl 1220.76016
[8] , , , , and , Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. Rev. Lett. 81 (1998) 5338-5341. | MR 1745983 | Zbl 1042.76525
[9] and , On a practical implementation of particle methods. Appl. Numer. Math. 56 (2006) 1418-1431. | MR 2245465 | Zbl 1103.65103
[10] and , Particle methods for dispersive equations. J. Comput. Phys. 171 (2001) 708-730. | MR 1848732 | Zbl 0991.65008
[11] and , A particle method for the KdV equation. J. Sci. Comput. 17 (2002) 491-499. | MR 1910746 | Zbl 1001.76079
[12] , Numerical study of slightly viscous flow. J. Fluid Mech. 57 (1973) 785-796. | MR 395483
[13] , and , A convergent finite difference scheme for the Camassa-Holm equation with general H1 initial data. SIAM J. Numer. Anal. 46 (2008) 1554-1579. | MR 2391006 | Zbl 1172.35310
[14] and , Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal. 32 (2000) 616-636 (electronic). | MR 1786160 | Zbl 0972.65058
[15] and , Vortex methods. Cambridge University Press, Cambridge (2000). | MR 1755095 | Zbl 0953.76001
[16] and , A particle method to solve transport-diffusion equations, Part 1 : the linear case. Tech. Report 115, Ecole Polytechnique, Palaiseau, France (1983). | Zbl 0678.35077
[17] and , A particle method to solve the Navier-Stokes system. Numer. Math. 57 (1990) 805-827. | MR 1065526 | Zbl 0707.76029
[18] and , The weighted particle method for convection-diffusion equations. I. The case of an isotropic viscosity. Math. Comput. 53 (1989) 485-507. | MR 983559 | Zbl 0676.65121
[19] and , The weighted particle method for convection-diffusion equations. II. The anisotropic case. Math. Comput. 53 (1989) 509-525. | MR 983560 | Zbl 0676.65122
[20] and , A deterministic approximation of diffusion equations using particles. SIAM J. Sci. Statist. Comput. 11 (1990) 293-310. | MR 1037516 | Zbl 0713.65090
[21] , and , High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89-112. | MR 1854647 | Zbl 0967.65098
[22] , Convergence of vortex methods, Vortex methods and vortex motion. SIAM, Philadelphia, PA (1991) 33-58. | MR 1095603
[23] , and , Averaged Template Matching Equations, EMMCVPR, Lecture Notes in Computer Science 2134. Springer (2001) 528-543. | Zbl 1001.68646
[24] and , Convergence of a finite difference scheme for the Camassa-Holm equation. SIAM J. Numer. Anal. 44 (2006) 1655-1680 (electronic). | MR 2257121 | Zbl 1122.76065
[25] and , A convergent numerical scheme for the Camassa-Holm equation based on multipeakons. Discrete Contin. Dyn. Syst. 14 (2006) 505-523. | MR 2171724 | Zbl 1111.35061
[26] and , 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. | MR 2103008
[27] and , Wave structure and nonlinear balances in a family of evolutionary PDEs. SIAM J. Appl. Dyn. Syst. 2 (2003) 323-380 (electronic). | MR 2031278 | Zbl 1088.76531
[28] and , Interaction dynamics of singular wave fronts, under “Recent Papers” at http://cnls.lanl.gov/~staley/.
[29] , , and , Soliton dynamics in computational anatomy. NeuroImage 23 (2004) S170-S178.
[30] , and , A two-dimensional version of the Camassa-Holm equation, Symmetry and perturbation theory. World Sci. Publ., Cala Gonone, River Edge, NJ (2001) 120-127. | MR 1875473 | Zbl 1061.35114
[31] , , and , Evolution of nonlinear internal waves in the east and south China Sea. J. Geophys. Res. 103 (1998) 7995-8008.
[32] and , Introduction to mechanics and symmetry, Texts in Applied Mathematics 17, 2nd edition. Springer-Verlag, New York (1999). | MR 1723696 | Zbl 0933.70003
[33] and , The accuracy of symplectic integrators. Nonlinearity 5 (1992) 541-562. | MR 1158385 | Zbl 0747.58032
[34] and , N-particle dynamics of the Euler equations for planar diffeomorfism. Dyn. Syst. 22 (2007) 269-290. | MR 2354966 | Zbl 1149.37038
[35] , An analysis of particle methods. Numerical methods in fluid dynamics (Como, 1983), Lecture Notes in Math. 1127. Springer, Berlin (1985) 243-324. | MR 802214 | Zbl 0598.76003
[36] , , and , Singularity tracking for Camassa-Holm and Prandtl's equations. Appl. Numer. Math. 56 (2006) 1108-1122. | MR 2234843 | Zbl 1096.76036
[37] , Symmetry in mechanics : A gentle, modern introduction. Birkhäuser Boston Inc., Boston, MA (2001). | MR 1816059 | Zbl 0970.70003
[38] and , A local discontinuous Galerkin method for the Camassa-Holm equation. SIAM J. Numer. Anal. 46 (2008) 1998-2021. | MR 2399405 | Zbl 1173.65063