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Structure-Preserving Wavelet Algorithms for the Nonlinear Dirac Model

  • Xu Qian (a1) (a2), Hao Fu (a1) and Songhe Song (a1) (a3)
Abstract
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.

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*Corresponding author. Email:shsong@nudt.edu.cn (S. H. Song)
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[1] J. Hong and C. Li , Multi-symplectic Runge-Kutta methods for nonlinear Dirac equations, J. Comput. Phys., 211 (2006), pp. 448472.

[2] J. Xu , S. Shao and H. Tang , Numerical methods for nonlinear Dirac equation, J. Comput. Phys., 245 (2013), pp. 131149.

[3] A. Alvarez , Linear Crank-Nicholsen scheme for nonlinear Dirac equations, J. Comput. Phys., 99 (1992), pp. 348350.

[4] A. Alvarez and B. Carreras , Interaction dynamics for the solitary waves of a nonlinear Dirac model, Phys. Lett. A, 86 (1981), pp. 327332.

[5] H. Wang and H. Z. Tang , An efficient adaptive mesh redistribution method for a nonlinear Dirac equation, J. Comput. Phys., 222 (2007), pp. 176193.

[6] J. De Frutos and J. M. Sanz-Serna , Split-step spectral schemes for nonlinear Dirac systems, J. Comput. Phys., 83 (1989), pp. 407423.

[7] S. Shao and H. Tang , Interaction for the solitary waves of a nonlinear Dirac model, Phys. Lett. A, 345 (2005), pp. 119128.

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

[10] H. Zhu , L. Tang , S. Song , Y. Tang and D. Wang , Symplectic wavelet collocation method for Hamiltonian wave equations, J. Comput. Phys., 229 (2010), pp. 25502572.

[11] H. Zhu , S. Song and Y. Chen , Multi-symplectic wavelet collocation method for Maxwell’s equations, Adv. Appl. Math. Mech., 3 (2011), pp. 663688.

[12] H. Zhu , S. Song and Y. Tang , Multi-symplectic wavelet collocation method for the nonlinear Schrödinger equation and the Camassa–Holm equation, Comput. Phys. Commun., 182 (2011), pp. 616627.

[13] X. Qian , Y. Chen and S. Song , Novel conservative methods for Schrödinger equations with variable coefficients over long time, Commun. Comput. Phys., 15 (2014), pp. 692711.

[14] R. Mclachlan and G. Quispel , Splitting methods, Acta Numer., 11 (2002), pp. 341434.

[15] B. Ryland , R. Mclachlan and J. Frank , On the multisymplecticity of partitioned Runge–Kutta and splitting methods, Int. J. Comput. Math., 84 (2007), pp. 847869.

[16] L. Kong , J. Hong , F. Fu and J. Chen , Symplectic structure-preserving integrators for the two-dimensional Gross–Pitaevskii equation for BEC, J. Comput. Appl. Math., 235 (2011), pp. 49374948.

[17] Y. Ma , L. Kong , J. Hong and Y. Cao , High-order compact splitting multisymplectic method for the coupled nonlinear Schrödinger equations, Comput. Math. Appl., 61 (2011), pp. 319333.

[18] Y. Chen , H. Zhu and S. Song , Multi-symplectic splitting method for two-dimensional nonlinear Schrödinger equation, Commun. Theor. Phys., 56 (2011), pp. 617622.

[19] X. Qian , S. Song and Y. Chen , A semi-explicit multi-symplectic splitting scheme for 3-coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 185 (2014), pp. 12551264.

[20] X. Qian , S. Song , E. Gao and W. Li , Explicit multi-symplectic method for the Zakharov–Kuznetsov equation, China Phys. B, 21 (2012), 070206.

[21] S. Reich , Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comput. Phys., 157 (2000), pp. 473499.

[24] J. Chen , M. Qin and Y. Tang , Symplectic and multi-symplectic methods for the nonlinear Schrödinger equation, Comput. Math. Appl., 43 (2002), pp. 10951106.

[25] Y. Wang , J. Jiang and W. Cai , Numerical analysis of a multi-symplectic scheme for the time-domain Maxwell's equations, J. Math. Phys., 52 (2011), 123701.

[26] A. Aydin and B. Karasözen , Symplectic and multi-symplectic methods for coupled nonlinear Schrödinger equations with periodic solutions, Comput. Phys. Commun., 177 (2007), pp. 566583.

[27] A. Aydin and B. Karasözen , Lobatto IIIA-IIIB discretization of the strongly coupled nonlinear Schrödinger equation, J. Comput. Appl. Math., 235 (2011), pp. 47704779.

[28] J. Frank , Conservation of wave action under multisymplectic discretizations, J. Phys. A Math. Gen., 39 (2006), pp. 54795493.

[29] J. Frank , B. Moore and S. Reich , Linear PDEs and numerical method that preserve a multisymplectic conservation law, SIAM J. Sci. Comput., 28 (2006), pp. 260277.

[30] B. Moore and S. Reich , Backward error analysis for multi-symplectic integration methods, Numer. Math., 95 (2003), pp. 625652.

[31] B. Moore , Conformal multi-symplectic integration methods for forced-damped semi-linear wave equations, Math. Comput. Simul., 80 (2009), pp. 2028.

[32] J. Hong and Y. Sun , Generating functions of multi-symplectic RK methods via DW Hamilton–Jacobi equations, Numer. Math., 110 (2008), pp. 491519.

[33] U. Ascher and R. Mclachlan , Multi symplectic box schemes and the Korteweg-de Vries equation, Appl. Numer. Math., 48 (2004), pp. 255269.

[34] A. Islas and C. Schober , On the preservation of phase space structure under multisymplectic discretization, J. Comput. Phys., 197 (2004), pp. 585609.

[35] Y. Miyatake , T. Yaguchi and T. Matsuo , Numerical integration of the Ostrovsky equationbased on its geometric structures, J. Comput. Phys., 231 (2012), pp. 45424559.

[36] G. Strang , On the construction and comparison of difference scheme, SIAM J. Numer. Anal., 5 (1968), pp. 506517.

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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
  • URL: /core/journals/advances-in-applied-mathematics-and-mechanics
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