Skip to main content
×
Home
    • Aa
    • Aa

High Order Energy-Preserving Method of the “Good” Boussinesq Equation

  • Chaolong Jiang (a1), Jianqiang Sun (a1), Xunfeng He (a1) and Lanlan Zhou (a1)
Abstract
Abstract

The fourth order average vector field (AVF) method is applied to solve the “Good” Boussinesq equation. The semi-discrete system of the “good” Boussinesq equation obtained by the pseudo-spectral method in spatial variable, which is a classical finite dimensional Hamiltonian system, is discretizated by the fourth order average vector field method. Thus, a new high order energy conservation scheme of the “good” Boussinesq equation is obtained. Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the “good” Boussinesq equation exactly and simulate evolution of different solitary waves well.

Copyright
Corresponding author
*Corresponding author. Email addresses: huitong_qiao@163.com (C.-L. Jiang), sunjq123@qq.com (J.-Q. Sun), 2597996867@qq.com (X.-F. He), 1473289657@qq.com (L.-L. Zhou)
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] A. Aydin , and B. Karasözen , Symplectic and multisymplectic Lobatto methods for the “good” Boussinesq equation, J. Math. Phys., Vol. 49 (2008), pp. 083509.

[3] T.J. Bridges , S. Reich , Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A., Vol. 284 (2001), pp. 184193.

[5] E. Celledoni , V. Grimm , R.I. McLachlan , et al., Preserving energy resp. dissipation in numerical PDEs using the “average vector field” method, J. Comput. Phys, Vol. 231 (2012), pp. 67706789.

[6] J.D. Frutos , T. Ortega and J.M. Sanz-Serna , Pseudo-spectral method for the “Good” Boussinesq equation, Math. Comp, Vol. 57 (1991), pp. 109122.

[8] W.P. Hu , and Z.C. Deng , Multi-symplectic method for generalized Boussinesq equation., Appl. Math. Mech.-Engl. ed., Vol. 29 (2008), pp. 927932.

[9] E. Hairer , C. Lubich and G. Wanner , Geometric Numerical Integration: Structure-Preserving Algorithm for Ordinary Differential Equatuins, Springer, Berlin, 2nd ed., 2006.

[12] V.S. Manoranjan , A.R. Mitchell and J.Ll. Morris , Numerical solutions of the “Good” Boussinesq equation., SIAM J. Sci. Comput., Vol. 5 (1984), pp. 946957.

[13] V.S. Manoranjan , T. Ortega and J.M Sanz-Serna , Soliton and antisoliton interaction in the “Good” Boussinesq equation, J. Math. Phys., Vol. 29 (1988), pp. 19641968.

[14] R.I. McLachlan , G.R.W. Quispel and N. Robidoux , Geometric integration using discrete gradients., Phil. Trans. R. Soc. A., Vol. 357 (1999), pp. 10211045.

[15] T. Ortega and J.M. Sanz-Serna , Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation., Numer. Math., Vol. 58 (1990), pp. 215229.

[17] G.R. Quispel , G.R.W. McLaren and D.I. McLaren , A new class of energy-preserving numerical integration methods., J. Phys. A: Math. Theor., Vol. 41 (2008), pp. 045206.

[18] J.M. Sanz-Serna , Symplectic Runge-Kutta and related methods: Recent result, Phys. D., Vol. 60 (1992), pp. 293302.

[20] H.E- Zoheiry , Numerical investigation for the solitary waves interaction of the “good” Boussinesq equation., Appl. Numer. Math, Vol. 45 (2003), pp. 161173.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 22 *
Loading metrics...

Abstract views

Total abstract views: 133 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st September 2017. This data will be updated every 24 hours.