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Hamiltonian Boundary Value Method for the Nonlinear Schrödinger Equation and the Korteweg-de Vries Equation

  • Mingzhan Song (a1), Xu Qian (a1), Hong Zhang (a1) and Songhe Song (a1)
Abstract

In this paper, we introduce the Hamiltonian boundary value method (HBVM) to solve nonlinear Hamiltonian PDEs. We use the idea of Fourier pseudospectral method in spatial direction, which leads to the finite-dimensional Hamiltonian system. The HBVM, which can preserve the Hamiltonian effectively, is applied in time direction. Then the nonlinear Schrödinger (NLS) equation and the Korteweg-de Vries (KdV) equation are taken as examples to show the validity of the proposed method. Numerical results confirm that the proposed method can simulate the propagation and collision of different solitons well. Meanwhile the corresponding errors in Hamiltonian and other intrinsic invariants are presented to show the good preservation property of the proposed method during long-time numerical calculation.

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Corresponding author
*Corresponding author. Email: smz161619@163.com (M. Song), qianxu@nudt.edu.cn (X. Qian), zhanghnudt@163.com (H. Zhang), shsong@nudt.edu.cn (S. H. Song)
References
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[1] McLachlan, R., Symplectic integration of Hamiltonian wave equations, Numerische Mathematik, 66(1) (1993), pp. 465492.
[2] Feng, K. and Qin, M. Z., Symplectic difference schemes for Hamiltonian systems, in Symplectic Geometric Algorithms for Hamiltonian Systems, Springer Berlin Heidelberg, (2010), pp. 187211.
[3] Dahlby, M., Owren, B. and Yaguchi, T., Preserving multiple first integrals by discrete gradients, J. Phys. A Math. Theor., 44(30) (2011), 305205.
[4] Quispel, G. R. W. and McLaren, D. I., A new class of energy-preserving numerical integration methods, J. Phys. A Math. Theor., 41(4) (2008), 045206.
[5] Brugnano, L., Iavernaro, F. and Trigiante, D., Hamiltonian boundary value methods (energy preserving discrete line integral methods), J. Numer. Anal. Industrial Appl. Math., 5(1-2) (2010), pp. 1737.
[6] Brugnano, L. and Iavernaro, F., Line integral methods which preserve all invariants of conservative problems, J. Comput. Appl. Math., 236(16) (2012), pp. 39053919.
[7] Bridges, T. J. and Reich, S., Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A, 284 (2001), pp. 184193.
[8] Zhu, H. J., Tang, L. Y., Song, S. H., Tang, Y. F. and Wang, D. S., Symplectic wavelet collocation method for Hamiltonian wave equations, J. Comput. Phys., 229 (2010), pp. 25502572.
[9] Qian, X., Song, S. H. and Gao, E., Explicit multi-symplectic method for the Zakharov-Kuznetsov equation, Chinese Phys. B, 21(7) (2012), pp. 4348.
[10] Li, H. C., Sun, J. Q. and Qin, M., Multi-symplectic method for the Zakharov-Kuznetsov equation, Adv. Appl. Math. Mech., 7(1) (2015), pp. 5873.
[11] Qian, X., Song, S. H. and Chen, Y. M., A semi-explicit multi-symplectic splitting scheme for a 3-coupled nonlinear Schrödinger equation, Comput. Phys. Commun., 185(4) (2014), pp. 12551264.
[12] Takaharu, Y., Matsuo, T. and Sugihara, M., The discrete variational derivative method based on discrete differential forms, J. Comput. Phys., 231(10) (2012), pp. 39633986.
[13] Miyatake, Y. and Matsuo, T., Conservative finite difference schemes for the Degasperis-Procesi equation, J. Comput. Appl. Math., 236(15) (2012), pp. 37283740.
[14] Hairer, E., Energy-preserving variant of collocation methods, J. Numer. Anal. Industrial Appl. Math., 5 (2010), pp. 7384.
[15] Miyatake, Y., An energy-preserving exponentially-fitted continuous stage Runge-Kutta method for Hamiltonian systems, BIT Numer. Math., 54(3) (2014), pp. 777799.
[16] Celledoni, E., Grimm, V., McLachlan, R. I., McLaren, D. I., O’Neale, D., Owren, B. and Quispel, G. R. W., Preserving energy resp. dissipation in numerical PDEs using the “Average Vector Field” method, J. Comput. Phys., 230(20) (2012), pp. 67706789.
[17] Zhang, H. and Song, S. H., Average vector field methods for the coupled Schrödinger KdV equations, Chinese Phys. B, 23(7) (2014), pp. 242250.
[18] Cai, J. X., Wang, Y. S. and Gong, Y. Z., Numerical analysis of AVF methods for three-dimensional time-domain Maxwell's equations, J. Sci. Comput., (2015), pp. 136.
[19] Karasozen, B. and Simsek, G., Energy preserving integration of bi-Hamiltonian partial differential equations, Appl. Math. Lett., 26(12) (2013), pp. 11251133.
[20] Brugnano, L., Caccia, G. F. and Iavernaro, F., Energy conservation issues in the numerical solution of the semilinear wave equation, Appl. Math. Comput., 270(C) (2015), pp. 842870.
[21] Brugnano, L. and Sun, Y. J., Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems, Numer. Algorithms, 65(3) (2014), pp. 611632.
[22] Chen, Y. M., Song, S. H. and Zhu, H. J., Explicit multi-symplectic splitting methods for the nonlinear Dirac equation, Adv. Appl. Math. Mech., 6(4) (2014), pp. 494514.
[23] Chen, Y. M., Song, S. H. and Zhu, H. J., Multi-symplectic methods for the Ito-type coupled KdV equation, Appl. Math. Comput., 218(9) (2012), pp. 55525561.
[24] Chen, J. B. and Qin, M. Z., Multi-symplectic fourier pseudospectral method for the nonlinear Schrödinger equation, Electronic Transactions on Numerical Analysis, 12 (2001), pp. 193204.
[25] Ascher, U. M. and McLachlan, R. I., On symplectic and multisymplectic schemes for the KdV equation, J. Sci. Comput., 25(1) (2005), pp. 83104.
[26] Wei, L. L., He, Y. N. and Zhang, X. D., Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Kdv equation, Adv. Appl. Math. Mech., 7(4) (2015), pp. 510527.
[27] Kieri, E., Kreiss, G. and Runborg, O., Coupling of Gaussian beam and finite difference solvers for semiclassical Schrödinger equations, Adv. Appl. Math. Mech., 7(6) (2015), pp. 687714.
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Advances in Applied Mathematics and Mechanics
  • ISSN: 2070-0733
  • EISSN: 2075-1354
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