A Particle Method for the b-Equation

DOI: 10.4236/jamp.2014.211111   PDF   HTML     2,263 Downloads   2,622 Views  

Abstract

In this paper, we apply the particle method to solve the numerical solution of a family of non-li-near Evolutionary Partial Differential Equations. It is called b-equation because of its bi-Hamiltonian structure. We introduce the particle method as an approximation of these equations in Lagrangian representation for simulating collisions between wave fronts. Several numerical examples will be set to illustrate the feasibility of the particle method.

Share and Cite:

Xing, Z. , Duan, Y. and Wu, H. (2014) A Particle Method for the b-Equation. Journal of Applied Mathematics and Physics, 2, 981-986. doi: 10.4236/jamp.2014.211111.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Camassa, R., Huang, J. and Lee, L. (2006) Integral and Integrable Algorithms for a Nonlinear Shallow-Water Wave Equation. Journal of Computational Physics, 216, 547-572. http://dx.doi.org/10.1016/j.jcp.2005.12.013
[2] Chertock, A., Du Toit, P. and Marsden, J. (2012) Integration of the EPDiff Equation by Particle Methods. ESAIM: Mathematical Modelling and Numerical Analysis, 46, 515-534. http://dx.doi.org/10.1051/m2an/2011054
[3] Chertock, A., Liu, J.G. and Pendleton, T. (2014) Elastic Collisions among Peakon Solutions for the Camassa-Holm Equation. Applied Numerical Mathematics, in press. http://dx.doi.org/10.1016/j.apnum.2014.01.001
[4] Matsuo, T. and Miyatake, Y. (2012) Conservative Finite Difference Schemes for Degasperis-Procesi Equation. Journal of Computational and Applied Mathematics, 236, 3728-3740. http://dx.doi.org/10.1016/j.cam.2011.09.004
[5] Camassa, R. and Lee, L. (2007) A Completely Integrable Particle Method for a Nonlinear Shallow-Water Wave Equation in Periodic Domains. Syst. Ser. A Math. Anal, 14, 1-5.
[6] Chertock, A., Liu, J.G. and Pendleton, T. (2012) Convergence of a Particle Method and Global Weak Solutions of a Family of Evolutionary PDEs. SIAM Journal on Numerical Analysis, 50, 1-21. http://dx.doi.org/10.1137/110831386
[7] Camassa, R., Huang, J. and Lee, L. (2005) On a Completely Integrable Numerical Scheme for a Nonlinear Shallow-Water Wave Equation. Journal of Nonlinear Mathematical Physics, 12, 146-162. http://dx.doi.org/10.2991/jnmp.2005.12.s1.13
[8] Holden, H. and Raynaud, X. (2006) Convergence of a Finite Difference Scheme for the Camassa-Holm Equation. SIAM Journal on Numerical Analysis, 44, 1655-1680. http://dx.doi.org/10.1137/040611975

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.