Free Access
Issue
ESAIM: M2AN
Volume 46, Number 3, May-June 2012
Special volume in honor of Professor David Gottlieb
Page(s) 515 - 534
DOI https://doi.org/10.1051/m2an/2011054
Published online 11 January 2012
  1. 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. [CrossRef]
  2. R. Artebrant and H.J. Schroll, Numerical simulation of Camassa-Holm peakons by adaptive upwinding. Appl. Numer. Math. 56 (2006) 695–711. [CrossRef]
  3. 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). [CrossRef]
  4. R. Camassa and D.D. Holm, An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71 (1993) 1661–1664. [CrossRef] [MathSciNet] [PubMed]
  5. R. Camassa, D.D. Holm and J.M. Hyman, A new integrable shallow water equation. Adv. Appl. Mech. 31 (1994) 1–33. [CrossRef]
  6. 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. [CrossRef]
  7. 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. [CrossRef]
  8. 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. [CrossRef] [MathSciNet]
  9. A. Chertock and A. Kurganov, On a practical implementation of particle methods. Appl. Numer. Math. 56 (2006) 1418–1431. [CrossRef]
  10. A. Chertock and D. Levy, Particle methods for dispersive equations. J. Comput. Phys. 171 (2001) 708–730. [CrossRef]
  11. A. Chertock and D. Levy, A particle method for the KdV equation. J. Sci. Comput. 17 (2002) 491–499. [CrossRef]
  12. A.J. Chorin, Numerical study of slightly viscous flow. J. Fluid Mech. 57 (1973) 785–796. [CrossRef]
  13. 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. [CrossRef] [MathSciNet]
  14. A. Cohen and B. Perthame, Optimal approximations of transport equations by particle and pseudoparticle methods. SIAM J. Math. Anal. 32 (2000) 616–636 (electronic). [CrossRef] [MathSciNet]
  15. G.-H. Cottet and P.D. Koumoutsakos, Vortex methods. Cambridge University Press, Cambridge (2000).
  16. 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).
  17. G.-H. Cottet and S. Mas-Gallic, A particle method to solve the Navier-Stokes system. Numer. Math. 57 (1990) 805–827. [CrossRef] [MathSciNet]
  18. 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.
  19. P. Degond and S. Mas-Gallic, The weighted particle method for convection-diffusion equations. II. The anisotropic case. Math. Comput. 53 (1989) 509–525.
  20. P. Degond and F.-J. Mustieles, A deterministic approximation of diffusion equations using particles. SIAM J. Sci. Statist. Comput. 11 (1990) 293–310. [CrossRef] [MathSciNet]
  21. S. Gottlieb, C.-W. Shu and E. Tadmor, High order time discretization methods with the strong stability property. SIAM Rev. 43 (2001) 89–112. [NASA ADS] [CrossRef] [MathSciNet]
  22. O.-H. Hald, Convergence of vortex methods, Vortex methods and vortex motion. SIAM, Philadelphia, PA (1991) 33–58.
  23. A.N. Hirani, J.E. Marsden and J. Arvo, Averaged Template Matching Equations, EMMCVPR, Lecture Notes in Computer Science 2134. Springer (2001) 528–543. [CrossRef]
  24. 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). [CrossRef] [MathSciNet]
  25. 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. [MathSciNet]
  26. 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.
  27. 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). [CrossRef]
  28. D.D. Holm and M.F. Staley, Interaction dynamics of singular wave fronts, under “Recent Papers” at http://cnls.lanl.gov/~staley/.
  29. D.D Holm, J.T. Ratnanather, A. Trouvé and L. Younes, Soliton dynamics in computational anatomy. NeuroImage 23 (2004) S170–S178. [CrossRef] [PubMed]
  30. 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.
  31. 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. [CrossRef]
  32. J.E. Marsden and T.S. Ratiu, Introduction to mechanics and symmetry, Texts in Applied Mathematics 17, 2nd edition. Springer-Verlag, New York (1999).
  33. R.I. McLachlan and P. Atela, The accuracy of symplectic integrators. Nonlinearity 5 (1992) 541–562. [CrossRef]
  34. R. McLachlan and S. Marsland, N-particle dynamics of the Euler equations for planar diffeomorfism. Dyn. Syst. 22 (2007) 269–290. [CrossRef] [MathSciNet]
  35. 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.
  36. 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. [CrossRef]
  37. S.F. Singer, Symmetry in mechanics : A gentle, modern introduction. Birkhäuser Boston Inc., Boston, MA (2001).
  38. Y. Xu and C.-W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation. SIAM J. Numer. Anal. 46 (2008) 1998–2021. [CrossRef] [MathSciNet]

Recommended for you