Volume 6, Issue 4
Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

Adv. Appl. Math. Mech., 6 (2014), pp. 494-514.

Published online: 2014-06

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• 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.

• Keywords

Nonlinear Dirac equation multi-symplectic method splitting method explicit method

65M06 65M70

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@Article{AAMM-6-494, author = {Yaming Chen, Songhe Song and Huajun Zhu}, title = {Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2014}, volume = {6}, number = {4}, pages = {494--514}, 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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2013.m278}, url = {http://global-sci.org/intro/article_detail/aamm/31.html} }
TY - JOUR T1 - Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation AU - Yaming Chen, Songhe Song & Huajun Zhu JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 494 EP - 514 PY - 2014 DA - 2014/06 SN - 6 DO - http://doi.org/10.4208/aamm.2013.m278 UR - https://global-sci.org/intro/article_detail/aamm/31.html KW - Nonlinear Dirac equation KW - multi-symplectic method KW - splitting method KW - explicit method AB -

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.

Yaming Chen, Songhe Song & Huajun Zhu. (1970). Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation. Advances in Applied Mathematics and Mechanics. 6 (4). 494-514. doi:10.4208/aamm.2013.m278
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