Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-12T03:53:07.825Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric Numerical Integration for Peakon b-Family Equations

Part of: Waves Partial differential equations, boundary value problems Partial differential equations, initial value and time-dependent initial-boundary value problems Numerical problems in dynamical systems Hyperbolic equations and systems

Published online by Cambridge University Press:  15 January 2016

Wenjun Cai ,
Yajuan Sun  and
Yushun Wang
Wenjun Cai
Affiliation:
Key Laboratory of Computational Geodynamics, University of Chinese Academy of Sciences, Beijing 100049, China Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University Nanjing 210023, China
Yajuan Sun*
Affiliation:
LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
Yushun Wang
Affiliation:
Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University Nanjing 210023, China
*
*Corresponding author. Email addresses:wenjuncai1@gmail.com (W. Cai), sunyj@lsec.cc.ac.cn (Y. Sun), wangyushun@njnu.edu.cn (Y. Wang)
Get access

Abstract

In this paper, we study the Camassa-Holm equation and the Degasperis-Procesi equation. The two equations are in the family of integrable peakon equations, and both have very rich geometric properties. Based on these geometric structures, we construct the geometric numerical integrators for simulating their soliton solutions. The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties, however they also have the significant difference, for example there exist the shock wave solutions for the Degasperis-Procesi equation. By using the symplectic Fourier pseudo-spectral integrator, we simulate the peakon solutions of the two equations. To illustrate the smooth solitons and shock wave solutions of the DP equation, we use the splitting technique and combine the composition methods. In the numerical experiments, comparisons of these two kinds of methods are presented in terms of accuracy, computational cost and invariants preservation.

Keywords

Symplectic integratorsplitting methodWENO schememultisymplectic integratorpeakonshockpeakon

MSC classification

Secondary: 35L65: Conservation laws 65M70: Spectral, collocation and related methods 65N06: Finite difference methods 65P10: Hamiltonian systems including symplectic integrators 74J40: Shocks and related discontinuities
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 1 , January 2016 , pp. 24 - 52
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Beals, R., Sattinger, D.H., and Szmigielski, J.. Multi-peakons and a theorem of Stieltjes. Inv. Prob., 15:L1–L4, 1999.Google Scholar
[2]Beals, R., Sattinger, D.H., and Szmigielski, J.. Multipeakons and the classical moment problem. Adv. Math., 154:229–257, 2000.Google Scholar
[3]Benjamin, T.B., Bona, J.L., and Mahony, J.. Model equations for long waves in nonlinear dispersive systems. Phil. R. Soc., 272:47–78, 1972.Google Scholar
[4]Bridges, T.J. and Reich, S.. Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations. Physica D, 152-153:491–504, 2001.CrossRefGoogle Scholar
[5]Camassa, R. and Holm, D.D.. A integrable shallow water equation with peaked solutions. Phys. Rev. Lett., 71:1661–1664, 1993.Google Scholar
[6]Camassa, R., Holm, D.D., and Hyman, J.M.. A new integrable shallow water equation. Adv. Appl. Mech., 31:1–33, 1994.Google Scholar
[7]Chen, J.B. and Qin, M.Z.. Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation. Electon. Trans. Numer. Anal., 12:193–204, 2001.Google Scholar
[8]Coclite, G.M., Karlsen, K., and Risebro, N.. Numerical schemes for computing discontinuous solutions of the Degasperis-Procesi equation. IMA J. Numer. Anal., 28:80–105, 2008.Google Scholar
[9]Coclite, G.M. and Karlsen, K.H.. On the well-posedness of the Degasperis-Procesi equation. J. Funct. Anal., 233:60–91, 2006.Google Scholar
[10]Coclite, G.M. and Karlsen, K.H.. On the uniqueness of discontinuous solutions to the Degasperis-Procesi equation. J. Differ. Equations, 234:142–160, 2007.Google Scholar
[11]Cohen, D., Owren, B., and Raynaud, X.. Multi-symplectic integration of the Camassa-Holm equation. J. Comput. Phys., 227:5492–5512, 2008.Google Scholar
[12]Constantin, A. and Lannes, D.. The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equation. Arch. Ration. Mech. Anal., 192:165–186, 2009.Google Scholar
[13]Degasperis, A., Holm, D.D., and Hone, A.H.W.. A new integrable equation with peakon solutions. Theor. Math. Phys., 133:1463–1474, 2002.Google Scholar
[14]Degasperis, A. and Procesi, M.. Asymptotic integrability In Symmetry and Perturbation Theory, pages 22–37. World Scientific Publishing, 1999.Google Scholar
[15]Escher, J., Liu, Y., and Yin, Z.. Global weak solutions and blow-up structure for the Degasperis-Procesi equation. J. Funct. Anal., 241:457–485, 2006.CrossRefGoogle Scholar
[16]Feng, B. and Liu, Y.. An operator splitting method for the Degasperis-Procesi equation. J. Comput. Phys., 228:7805–7820, 2009.Google Scholar
[17]Gottlieb, S. and Shu, C.-W.. Total variation diminishing Runge-Kutta schemes. Math. Comput., 67:73–85, 1998.Google Scholar
[18]Gottlieb, S., Shu, C.-W., and Tadmor, E.. Strong stability preserving high order time discretization methods. SIAM Rev., 43:89–112, 2001.Google Scholar
[19]Hairer, E., Lubich, C., and Wanner, G.. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer-Verlag, Berlin, second edition, 2006.Google Scholar
[20]Hoel, H.. A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation. Electron. J. Differential Equations, 2007:1–22, 2007.Google Scholar
[21]Jiang, G. and Shu, C.-W.. Efficient implementation of Weighted ENO schemes. J. Comput. Phys., 126:202–228, 1996.CrossRefGoogle Scholar
[22]Johnson, R.S.. Camassa-Holm, Korteweg-de Vries and related models for water. J. Fluid Mech., 455:63–82, 2002.Google Scholar
[23]Kalisch, H. and Lenells, J.. Numerical study of traveling-wave solutions for the Camassa-Holm equation. Chaos Soliton Fract., 25:287–298, 2005.Google Scholar
[24]Kalisch, H. and Raynaud, X.. Convergence of a spectral projection of the Camassa-Holm equation. Numer. Meth. Part. D. E., 22:1197–1215, 2006.Google Scholar
[25]Liu, H., Huang, Y., and Yi, N.. A conservative discontinuous Galerkin method for the Degasperis-Procesi equation. Methods Appl. Anal., 21:83–106, 2014.Google Scholar
[26]Liu, Y. and Yin, Z.. Global existence and blow-up phenomena for the Degasperis-Procesi equation. Comm. Math. Phys., 267:801–820, 2006.Google Scholar
[27]Lundmark, H.. Formation and dynamics of shock waves in the Degasperis-Procesi equation. J. Nonlinear. Sci, 17:169–198, 2007.Google Scholar
[28]Lundmark, H. and Szmigielski, J.. Degasperis-Procesi peakons and the discrete cubic string. Int. Math. Res. Pap., 2005:53–116, 2003.CrossRefGoogle Scholar
[29]Lundmark, H. and Szmigielski, J.. Multi-peakon solutions of the Degasperis-Procesi equation. Inv. Prob., 19:1241–1245, 2003.Google Scholar
[30]Matsuno, Y.. Multisoliton solutions of the Degasperis-Procesi equation. Inv. Prob., 21:2085–2101, 2005.Google Scholar
[31]Matsuno, Y.. Multisoliton solutions of the Degasperis-Procesi equation and their peakon limit. Inv. Prob., 21:1553–1570, 2005.CrossRefGoogle Scholar
[32]Miyatake, Y. and Matsuo, T.. Conservative finite difference schemes for the Degasperis-Procesi equation. J. Comput. Appl. Math., 236:3728–3740, 2012.Google Scholar
[33]Olver, P.J.. On the Hamiltonian structure of evolution equations. Math. Proc. Camb. Phil. Soc. 88:71–88, 1980.Google Scholar
[34]Shen, J., Tang, T. and Wang, L.-L.. Spectral Methods: Algorithms, Analysis and Applications. Springer, 2011.Google Scholar
[35]Shu, C.-W.. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, pages 325–432. Springer Berlin Heidelberg, 1998.Google Scholar
[36]Shu, C.-W.. Total-Variation-Diminishing time discretizations. SIAM J. Sci. Stat. Comput., 9:1073–1084, 1998.Google Scholar
[37]Shu, C.-W. and Osher, S.. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys., 77:439–471, 1998.Google Scholar
[38]Sun, Y.J. and Qin, M.Z.. A multi-symplectic scheme for RLW equation. J. Comput. Math., 22:611–621, 2004.Google Scholar
[39]Trefethen, L.N.. Spectral Methods in MATLAB. SIAM, Philadelphia, 2000.Google Scholar
[40]Xia, Y.. Fourier spectral methods for Degasperis-Procesi equation with discontinuous solutions. J. Sci. Comput., 61:584–603, 2014.Google Scholar
[41]Xu, Y. and Shu, C.-W.. Local discontinuous Galerkin methods for the Degasperis-Procesi equation. Commun. Comput. Phys., 10:474–508, 2011.Google Scholar
[42]Yoshida, H.. Construction of higher order symplectic integrators. Phys. Lett. A, 150:262–268, 1990.Google Scholar
[43]Yoshida, H.. Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A, 146:319–323, 1990.Google Scholar
[44]Yu, C.H. and Sheu, Tony W.H.. A dispersively accurate compact finite difference method for the Degasperis-Procesi equation. J. Comput. Phys., 236:493–512, 2013.Google Scholar