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A conservative local discontinuous Galerkin method for the solution of nonlinear Schrödinger equation in two dimensions

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Abstract

In this study, we present a conservative local discontinuous Galerkin (LDG) method for numerically solving the two-dimensional nonlinear Schr¨odinger (NLS) equation. The NLS equation is rewritten as a firstorder system and then we construct the LDG formulation with appropriate numerical flux. The mass and energy conserving laws for the semi-discrete formulation can be proved based on different choices of numerical fluxes such as the central, alternative and upwind-based flux. We will propose two kinds of time discretization methods for the semi-discrete formulation. One is based on Crank-Nicolson method and can be proved to preserve the discrete mass and energy conservation. The other one is Krylov implicit integration factor (IIF) method which demands much less computational effort. Various numerical experiments are presented to demonstrate the conservation law of mass and energy, the optimal rates of convergence, and the blow-up phenomenon.

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Acknowledgments

This work was supported by the Foundation of Liaoning Educational Committee (Grant No. L201604) and China Scholarship Council, National Natural Science Foundation of China (Grant Nos. 11571002, 11171281 and 11671044), the Science Foundation of China Academy of Engineering Physics (Grant No. 2015B0101021) and the Defense Industrial Technology Development Program (Grant No. B1520133015). The authors thank the referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang, 110034, China

    RongPei Zhang

  2. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    XiJun Yu

  3. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, China

    MingJun Li

  4. School of Applied Science, Beijing Information Science and Technology University, Beijing, Beijing, China

    XiangGui Li

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  1. RongPei Zhang
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  2. XiJun Yu
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  4. XiangGui Li
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Correspondence to MingJun Li.

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Zhang, R., Yu, X., Li, M. et al. A conservative local discontinuous Galerkin method for the solution of nonlinear Schrödinger equation in two dimensions. Sci. China Math. 60, 2515–2530 (2017). https://doi.org/10.1007/s11425-016-9118-x

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  • DOI: https://doi.org/10.1007/s11425-016-9118-x

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