Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T16:31:52.110Z Has data issue: false hasContentIssue false
  • English
  • Français

Multi-Symplectic Fourier Pseudospectral Method for the Kawahara Equation

Published online by Cambridge University Press:  03 June 2015

Yuezheng Gong ,
Jiaxiang Cai  and
Yushun Wang
Yuezheng Gong*
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
Jiaxiang Cai*
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
Yushun Wang*
Affiliation:
Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing 210023, P.R. China
*
Email:[email protected]
Email:[email protected]
Corresponding author.Email:[email protected]
Get access

Abstract

In this paper, we derive a multi-symplectic Fourier pseudospectral scheme for the Kawahara equation with special attention to the relationship between the spectral differentiation matrix and discrete Fourier transform. The relationship is crucial for implementing the scheme efficiently. By using the relationship, we can apply the Fast Fourier transform to solve the Kawahara equation. The effectiveness of the proposed methods will be demonstrated by a number of numerical examples. The numerical results also confirm that the global energy and momentum are well preserved.

Keywords

65M0665M7065T5065Z0570H15Kawahara equationMulti-symplecticityFourier pseudospectral methodFFT.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 35 - 55
Copyright
Copyright © Global Science Press Limited 2014

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]Bridges, T.J. and Derks, G., Linear instability of solitary wave solution of the Kawahara equation and its generalizations, SIAM J. Math. Anal., 33(2002), 1356–1378.Google Scholar
[2]Kawahara, R., Oscillatory solitary waves in dispersive media, J. Phys. Soc. Japan, 33(1972), 260–264.Google Scholar
[3]Olver, PJ., Hamiltonian perturbation theory and water waves, Contemp. Math., 28(1984), 231–249.Google Scholar
[4]Craig, W. and Groves, M.D., Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19(1994), 367–389.CrossRefGoogle Scholar
[5]Tao, S.P., Cui, S.B., Existence and uniqueness of solutions to nonlinear Kawahara equations, Chin. Ann. Math. Ser. A, 23(2002), 221–228.Google Scholar
[6]Yuan, J.M., Shen, J. and Wu, J.H., A Dual-Petrov-Garlerkin method for the Kawahara-type equation, J. Sci. Comput., 34(2008), 48–63.Google Scholar
[7]Hu, W.P. and Deng, Z.C., Multi-symplectic method for generalized fifth-order KdV equation, Chin. Phys. B, 17(2008), 3923–3929.Google Scholar
[8]Bridges, T.J., Multi-symplectic structures and wave propagation, Math. Proc. Camb. Phil. Soc., 121(1997), 147–190.Google Scholar
[9]Bridges, T.J. and Reich, S., Multi-symplectic Integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284(2001), 184–193.Google Scholar
[10]Reich, S., Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157(2000), 473–499.Google Scholar
[11]Bridges, T.J. and Reich, S., Numerical methods for Hamiltonian PDEs, J. Phys. A: Math. Gen., 39(2006), 5287–5320.CrossRefGoogle Scholar
[12]Bridges, T.J. and Reich, S., Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Phys. D, 152-153(2001), 491–504.Google Scholar
[13]Chen, J.B. and Qin, M.Z., Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation, Electr. Trans. Numer. Anal, 12(2001), 193–204.Google Scholar
[14]Chen, Y.M., Song, S.H. and Zhu, H.J., Multi-symplectic methods for the Ito-type coupled KdV equation, Appl. Math. Comput., 218(2012), 5552–5561.Google Scholar
[15]Zhu, H.J., Song, S.H. and Chen, Y.M., Multi-symplectic wavelet collocation method for Maxwell’s equations, Appl. Math. Mech., 3(2011), 663–688.Google Scholar
[16]Moore, B. and Reich, S., Backward error analysis for multi-symplectic integration methods, Numer. Math., 95(2003), 625–652.Google Scholar
[17]Wang, Y.S., Wang, B. and Chen, X., Multisymplectic Euler box scheme for the KdV equation, Chin. Phys. Lett., 24(2007), 312–314.Google Scholar
[18]Chen, Y., Sun, Y.J. and Tang, Y.F., Energy-preserving numerical methods for LandauCLifshitz equation, J. Phys. A: Math. Theor., 44(29)(2011): 295207.Google Scholar
[19]Cai, J.X., A multisymplectic explicit scheme for the modified regularized long-wave equation, J. Comput. Appl. Math., 234(2010), 899–905.Google Scholar
[20]Ryland, B.N., McLachlan, B.I. and Frank, J., On multisymplecticity of partitioned Runge-Kutta and splitting methods, Int. J. Comput. Math., 84(2007), 847–869.Google Scholar
[21]Kong, L.H., Hong, J.L. and Zang, J.J., Splitting multi-symplectic integrators for Maxwell’s equation, J. Comput. Phys., 229(2010), 4259–4278.Google Scholar
[22]Kong, L.H., Zhang, J.J., Cao, Y., Duan, Y.L. and Huang, H., Semi-explicit symplectic partitioned Runge-Kutta Fourier pseudo-spectral scheme for Klein-Gordon-Schrodinger equations, Comput. Phys.Commun., 181(2010), 1369–1377.CrossRefGoogle Scholar
[23]Wang, J., A note on multi-symplectic fourier pseudospectral discretization for the nonlinear Schrodinger equation, Appl. Math. Comput., 191(2007), 31–41.Google Scholar
[24]Chen, J.B., Symplectic And Multisymplectic Fourier Pseudospectral Discretizations for the Klein-Gordon Equation, Lett. Math. Phys., 75(2006), 293–305.Google Scholar
[25]Lv, Z.Q., Xue, M., and Wang, Y.S., A new multi-symplectic scheme for the KdV equation, Chin. Phys. Lett., 28(6)(2011):060205.Google Scholar
[26]Benjamin, T.B., The stability of solitary waves, Proc. Roy. Soc. London A, 23(1972), 153–183.Google Scholar