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Linear High-Order Energy-Preserving Schemes for the Nonlinear Schrödinger Equation with Wave Operator Using the Scalar Auxiliary Variable Approach

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Abstract

In this paper, we develop two classes of linear high-order conservative numerical schemes for the nonlinear Schrödinger equation with wave operator. Based on the method of order reduction in time and the scalar auxiliary variable technique, we transform the original model into an equivalent system, where the energy is modified as a quadratic form. To construct linear high-order conservative schemes, we first adopt the extrapolation strategy to derive a linearized PDE system, which approximates the transformed model with high precision and inherits the modified energy conservation law. Then we employ the symplectic Runge–Kutta method in time to arrive at a class of linear high-order energy-preserving schemes. This numerical strategy presents a paradigm for developing arbitrarily high-order linear structure-preserving algorithms which could be implemented simply. In order to complement the new linear schemes, the prediction-correction method is presented to obtain another class of energy-preserving algorithms. Furthermore, the trigonometric pseudo-spectral method is applied for the spatial discretization to match the order of accuracy in time. We provide ample numerical results to confirm the convergence, accuracy and conservation property of the proposed schemes.

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, HeFei, 230009, Anhui, People’s Republic of China

    Xin Li

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, Jiangsu, People’s Republic of China

    Yuezheng Gong & Luming Zhang

  3. Jiangsu Key Lab for NSLSCS, Nanjing, 210023, Jiangsu, People’s Republic of China

    Yuezheng Gong

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  1. Xin Li
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  2. Yuezheng Gong
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  3. Luming Zhang
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Correspondence to Yuezheng Gong.

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Yuezheng Gong: He is supported by the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems (202002), a grant BK20180413 from the Nature Science Foundation of Jiangsu Province and a grant 11801269 from the National Nature Science Foundation of China.

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Li, X., Gong, Y. & Zhang, L. Linear High-Order Energy-Preserving Schemes for the Nonlinear Schrödinger Equation with Wave Operator Using the Scalar Auxiliary Variable Approach. J Sci Comput 88, 20 (2021). https://doi.org/10.1007/s10915-021-01533-9

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  • DOI: https://doi.org/10.1007/s10915-021-01533-9

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