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Energy Conserving Local Discontinuous Galerkin Methods for the Nonlinear Schrödinger Equation with Wave Operator

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Abstract

In this paper, we present a fully discrete scheme by discretizing the space with the local discontinuous Galerkin method and the time with the Crank–Nicholson scheme to simulate the multi-dimensional Schrödinger equation with wave operator. The scheme can preserve the energy conservation which is an important property of the nonlinear Schrödinger equation with wave operator. The energy conservation is also a crucial property for long time simulations which will be demonstrated in the numerical experiment. The optimal error estimates of the semi-discrete scheme can be obtained for the linear case. Some numerical experiments in multi-dimensional spaces are shown to demonstrate the accuracy and capability of this scheme.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Li Guo & Yan Xu

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  1. Li Guo
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  2. Yan Xu
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Correspondence to Yan Xu.

Additional information

Research supported by NSFC Grant Nos. 11371342, 11031007, Fok Ying Tung Education Foundation No. 131003.

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Guo, L., Xu, Y. Energy Conserving Local Discontinuous Galerkin Methods for the Nonlinear Schrödinger Equation with Wave Operator. J Sci Comput 65, 622–647 (2015). https://doi.org/10.1007/s10915-014-9977-z

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  • DOI: https://doi.org/10.1007/s10915-014-9977-z

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