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Structure-Preserving Wavelet Algorithms for the Nonlinear Dirac Model

Part of: Numerical approximation and computational geometry Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  18 January 2017

Xu Qian ,
Hao Fu  and
Songhe Song
Xu Qian
Affiliation:
Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410072, China Academy of Ocean Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, China
Hao Fu
Affiliation:
Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410072, China
Songhe Song*
Affiliation:
Department of Mathematics and Systems Science, National University of Defense Technology, Changsha, Hunan 410072, China State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, Hunan 410073, China
*
*Corresponding author. Email:shsong@nudt.edu.cn (S. H. Song)
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Abstract

The nonlinear Dirac equation is an important model in quantum physics with a set of conservation laws and a multi-symplectic formulation. In this paper, we propose energy-preserving and multi-symplectic wavelet algorithms for this model. Meanwhile, we evidently improve the efficiency of these algorithms in computations via splitting technique and explicit strategy. Numerical experiments are conducted during long-term simulations to show the excellent performances of the proposed algorithms and verify our theoretical analysis.

Keywords

Structure-preservingwavelet collocationconservation lawsnonlinear Dirac equation

MSC classification

Secondary: 35Q41: Time-dependent Schrödinger equations, Dirac equations 65D30: Numerical integration
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 4 , August 2017 , pp. 964 - 989
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Hong, J. and Li, C., Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations, J. Comput. Phys., 211 (2006), pp. 448–472.Google Scholar
[2] Xu, J., Shao, S. and Tang, H., Numerical methods for nonlinear Dirac equation, J. Comput. Phys., 245 (2013), pp. 131–149.CrossRefGoogle Scholar
[3] Alvarez, A., Linear Crank-Nicholsen scheme for nonlinear Dirac equations, J. Comput. Phys., 99 (1992), pp. 348–350.Google Scholar
[4] Alvarez, A. and Carreras, B., Interaction dynamics for the solitary waves of a nonlinear Dirac model, Phys. Lett. A, 86 (1981), pp. 327–332.CrossRefGoogle Scholar
[5] Wang, H. and Tang, H. Z., An efficient adaptive mesh redistribution method for a nonlinear Dirac equation, J. Comput. Phys., 222 (2007), pp. 176–193.Google Scholar
[6] De Frutos, J. and Sanz-Serna, J. M., Split-step spectral schemes for nonlinear Dirac systems, J. Comput. Phys., 83 (1989), pp. 407–423.CrossRefGoogle Scholar
[7] Shao, S. and Tang, H., Interaction for the solitary waves of a nonlinear Dirac model, Phys. Lett. A, 345 (2005), pp. 119–128.Google Scholar
[8] Shao, S. and Tang, H., Interaction of solitary waves with a phase shift in a nonlinear Dirac model, Commun. Comput. Phys., 3 (2008), pp. 950–967.Google Scholar
[9] Bridges, T. and Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), pp. 184–193.Google Scholar
[10] 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. 2550–2572.CrossRefGoogle Scholar
[11] Zhu, H., Song, S. and Chen, Y., Multi-symplectic wavelet collocation method for Maxwell’s equations, Adv. Appl. Math. Mech., 3 (2011), pp. 663–688.Google Scholar
[12] Zhu, H., Song, S. and Tang, Y., Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa–Holm equation, Comput. Phys. Commun., 182 (2011), pp. 616–627.Google Scholar
[13] Qian, X., Chen, Y. and Song, S., Novel conservative methods for Schrödinger equations with variable coefficients over long time, Commun. Comput. Phys., 15 (2014), pp. 692–711.CrossRefGoogle Scholar
[14] Mclachlan, R. and Quispel, G., Splitting methods, Acta Numer., 11 (2002), pp. 341–434.CrossRefGoogle Scholar
[15] Ryland, B., Mclachlan, R. and Frank, J., On the multisymplecticity of partitioned Runge–Kutta and splitting methods, Int. J. Comput. Math., 84 (2007), pp. 847–869.CrossRefGoogle Scholar
[16] Kong, L., Hong, J., Fu, F. and Chen, J., Symplectic structure-preserving integrators for the two-dimensional Gross–Pitaevskii equation for BEC, J. Comput. Appl. Math., 235 (2011), pp. 4937–4948.Google Scholar
[17] Ma, Y., Kong, L., Hong, J. and Cao, Y., High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations, Comput. Math. Appl., 61 (2011), pp. 319–333.CrossRefGoogle Scholar
[18] Chen, Y., Zhu, H. and Song, S., Multi-symplectic splitting method for two-dimensional nonlinear Schrödinger equation, Commun. Theor. Phys., 56 (2011), pp. 617–622.Google Scholar
[19] Qian, X., Song, S. and Chen, Y., A semi-explicit multi-symplectic splitting scheme for 3-coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 185 (2014), pp. 1255–1264.CrossRefGoogle Scholar
[20] Qian, X., Song, S., Gao, E. and Li, W., Explicit multi-symplectic method for the Zakharov–Kuznetsov equation, China Phys. B, 21 (2012), 070206.Google Scholar
[21] Reich, S., Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), pp. 473–499.Google Scholar
[22] Bridges, T. and Reich, S., Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Phys. D, 152 (2001), pp. 491–504.Google Scholar
[23] Chen, J. and Qin, M., Multi-symplectic Fourier pseudospectral method for the nonlinear Schrödinger equation, Electron. Trans. Numer. Anal., 12 (2001), pp. 193–204.Google Scholar
[24] Chen, J., Qin, M. and Tang, Y., Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 43 (2002), pp. 1095–1106.Google Scholar
[25] Wang, Y., Jiang, J. and Cai, W., Numerical analysis of a multi-symplectic scheme for the time-domain Maxwell's equations, J. Math. Phys., 52 (2011), 123701.CrossRefGoogle Scholar
[26] Aydin, A. and Karasözen, B., Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions, Comput. Phys. Commun., 177 (2007), pp. 566–583.CrossRefGoogle Scholar
[27] Aydin, A. and Karasözen, B., Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schrödinger equation, J. Comput. Appl. Math., 235 (2011), pp. 4770–4779.CrossRefGoogle Scholar
[28] Frank, J., Conservation of wave action under multisymplectic discretizations, J. Phys. A Math. Gen., 39 (2006), pp. 5479–5493.Google Scholar
[29] Frank, J., Moore, B. and Reich, S., Linear PDEs and numerical method that preserve a multisymplectic conservation law, SIAM J. Sci. Comput., 28 (2006), pp. 260–277.CrossRefGoogle Scholar
[30] Moore, B. and Reich, S., Backward error analysis for multi-symplectic integration methods, Numer. Math., 95 (2003), pp. 625–652.Google Scholar
[31] Moore, B., Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations, Math. Comput. Simul., 80 (2009), pp. 20–28.CrossRefGoogle Scholar
[32] Hong, J. and Sun, Y., Generating functions of multi-symplectic RK methods via DW Hamilton–Jacobi equations, Numer. Math., 110 (2008), pp. 491–519.CrossRefGoogle Scholar
[33] Ascher, U. and Mclachlan, R., Multi symplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48 (2004), pp. 255–269.CrossRefGoogle Scholar
[34] Islas, A. and Schober, C., On the preservation of phase space structure under multisymplectic discretization, J. Comput. Phys., 197 (2004), pp. 585–609.Google Scholar
[35] Miyatake, Y., Yaguchi, T. and Matsuo, T., Numerical integration of the Ostrovsky equationbased on its geometric structures, J. Comput. Phys., 231 (2012), pp. 4542–4559.CrossRefGoogle Scholar
[36] Strang, G., On the construction and comparison of difference scheme, SIAM J. Numer. Anal., 5 (1968), pp. 506–517.Google Scholar