Hostname: page-component-848d4c4894-8kt4b Total loading time: 0 Render date: 2024-06-18T13:35:23.876Z Has data issue: false hasContentIssue false

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

Published online by Cambridge University Press:  03 June 2015

Yaming Chen*
Affiliation:
Department of Mathematics and System Science and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China
Songhe Song*
Affiliation:
Department of Mathematics and System Science and State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha 410073, China
Huajun Zhu*
Affiliation:
State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000, China
*
Corresponding author. Email: shsong@nudt.edu.cn
Get access

Abstract

In this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.

Type
Research Article
Copyright
Copyright © Global-Science Press 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] Hong, J. and Li, C., Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations, J. Comput. Phys., 211 (2006), pp. 448472.CrossRefGoogle Scholar
[2] de Frutos, J. and Sanz-Serna, J., Split-step spectral schemes for nonlinear Dirac systems, J. Comput. Phys., 83 (1989), pp. 407423.CrossRefGoogle Scholar
[3] Alvarez, A., Linear Crank-Nicholsen scheme for nonlinear Dirac equations, J. Comput. Phys., 99 (1992), pp. 348350.CrossRefGoogle Scholar
[4] Alvarez, A., Kuo, P. and Vazquez, L., The numerical study of a nonlinear one-dimensional Dirac equation, Appl. Math. Comput., 13 (1983), pp. 115.Google Scholar
[5] Alvarez, A. and Carreras, B., Interaction dynamics for the solitary waves of a nonlinear Dirac model, Phys. Lett. A, 86 (1981), pp. 327332.Google Scholar
[6] Wang, H., An efficient adaptive mesh redistribution method for a nonlinear Dirac equation J. Comput. Phys., 222 (2006), pp. 176193.CrossRefGoogle Scholar
[7] Shao, S. and Tang, H., Interaction for the solitary waves of a nonlinear Dirac model, Phys. Lett. A, 345 (2005), pp. 119128.Google Scholar
[8] Shao, S. and Tang, H., Higher-order accurate Runge-Kutta discontinuous Galerkin methods for a nonlinear Dirac model, DCDS-B, 6 (2006), pp. 623640.Google Scholar
[9] Shao, S. and Tang, H., Interaction for solitary waves with a phase difference in a nonlinear Dirac model, Commun. Comput. Phys., 3 (2008), pp. 950967.Google Scholar
[10] de la Hoz, F. and Vadillo, F., An integrating factor for the nonlinear Dirac equations, Comput. Phys. Commun., 181 (2010), pp. 11951203.CrossRefGoogle Scholar
[11] Bridges, T. J. and Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), pp. 184193.CrossRefGoogle Scholar
[12] Feng, K. and Qin, M., Hamiltonian algorithms for Hamitonian systems and a comparative numerical study, Comput. Phys. Commun., 65 (1991), pp. 173187.Google Scholar
[13] Guan, H., Jiao, Y., Liu, J. and Tang, Y., Explicit symplectic methods for the nonlinear Schrödinger equation, Commun. Comput. Phys., 6 (2009), pp. 639654.Google Scholar
[14] Feng, Q., Huang, J., Nie, N., Shang, Z. and Tang, Y., Implementing arbitrarily high-order symplectic methods via Krylov deferred correction technique, Int. J. Model. Simul. Sci. Comput., 1 (2010), pp. 277301.CrossRefGoogle Scholar
[15] Zhang, R., Huang, J., Tang, Y. and Vszquez, L., Revertible and Symplectic Methods for the Ablowitz-Ladik Discrete Nonlinear Schrodinger Equation, in GCMS11 Proceedings of the 2011 Grand Challenges on Modeling and Simulation Conference, pp. 297306, Society for Modeling and Simulation International, Vista, CA, 2011.Google Scholar
[16] Bridges, T. J. and Reich, S., Numerical methods for Hamiltonian PDEs, J. Phys. A Math. Gen., 39 (2006), pp. 52875320.Google Scholar
[17] Huang, L., Jiao, Y. and Liang, D., Multi-symplectic scheme for the coupled Schrodinger Boussinesq equations, Chinese Phys. B, 22 (2013), 070201.CrossRefGoogle Scholar
[18] Reich, S., Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), pp. 473499.CrossRefGoogle Scholar
[19] Bridges, T. J. and Reich, S., Multi-symplectic spectral discretizations for the Zakharov- Kuznetsov and shallow water equations, Phys. D, 152-153 (2001), pp. 491504.CrossRefGoogle Scholar
[20] Chen, J. and Qin, M., Multi-symplectic Fourier pseudospectral method for the nonlinear Schrodinger equation, Electr. Numer. Anal., 12 (2001), pp. 193204.Google Scholar
[21] Chen, Y., Song, S. and Zhu, H., The multi-symplectic Fourier pseudospectral method for solving two-dimensional Hamiltonian PDEs, J. Comput. Appl. Math., 236 (2011), pp. 13541369.Google Scholar
[22] Zhu, H., Song, S. and Tang, Y., Multi-symplectic wavelet collocation method for the nonlinear Schrodinger equation and the Camassa-Holm equation, Comput. Phys. Commun., 182 (2011), pp. 616627.CrossRefGoogle Scholar
[23] Zhu, H., Chen, Y., Song, S. and Hu, H., Symplectic and multi-symplectic wavelet collocation method for the two-dimensional nonlinear Schrodinger equation, Appl. Numer. Math., 61 (2011), pp. 308321.CrossRefGoogle Scholar
[24] Zhu, H., Song, S. and Chen, Y., Multi-symplectic wavelet collocation method for Maxwell’s equations, Adv. Appl. Math. Mech., 3 (2011), pp. 663688.CrossRefGoogle Scholar
[25] Chen, Y., Song, S. and Zhu, H., Multi-symplectic methods for the Ito-type coupled KdV equation, Appl. Math. Comput., 218 (2012), pp. 55525561.Google Scholar
[26] Ryland, B. N., Mclachlan, R. I. and Frank, J., On multisymplecticity of partitioned Runge- Kutta and splitting methods, Int. J. Comput. Math., 84 (2007), pp. 847869.CrossRefGoogle Scholar
[27] Hong, J. and Kong, L., Novel multi-symplectic integrators for nonlinear fourth-order Schrodinger eqaution with trapped term, Commun. Comput. Phys., 7 (2010), pp. 613630.Google Scholar
[28] Kong, L., Hong, J. and Zang, J., Splitting multi-symplectic integrators for Maxwell’s equation, J. Comput. Phys., 229 (2010), pp. 42594278.CrossRefGoogle Scholar
[29] Chen, Y., Zhu, H. and Song, S., Multi-symplectic splitting method for the coupled nonlinear Schrodinger equation, Comput. Phys. Commun., 181 (2010), pp. 12311241.CrossRefGoogle Scholar
[30] Chen, Y., Zhu, H. and Song, S., Multi-symplectic splitting method for two-dimensional nonlinear Schrodinger equation, Commun. Theor. Phys., 56 (2011), pp. 617622.CrossRefGoogle Scholar
[31] Hong, J., Jiang, S., Li, C. and Liu, H., Explicit multi-symplectic methods for Hamiltonian wave equations, Commun. Comput. Phys., 2 (2007), pp. 662683.Google Scholar
[32] Hong, J., Jiang, S. and Li, C., Explicit multi-symplectic methods for Klein-Gordon-Schrodinger equations, J. Comput. Phys., 228 (2009), pp. 35173532.CrossRefGoogle Scholar
[33] Hong, J. and Li, C., Some properties of multi-symplectic Runge-Kutta methods for Dirac equations, Research Report of ICMSEC, 2004.Google Scholar
[34] Zhu, H., Tang, L., Song, S., Tang, Y. and Wang, D., Symplectic wavelet collocation method for Hamiltonian wave equations, J. Comput. Phys., 229 (2010), pp. 25502572.Google Scholar
[35] Ma, Y., Kong, L., Hong, J. and Cao, Y., High-order compact splitting multisymplectic method for the coupled nonlinear Schrodinger equations, Comput. Math. Appl., 61 (2011), pp. 319333.CrossRefGoogle Scholar