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High-Order Linearly Implicit Structure-Preserving Exponential Integrators for the Nonlinear Schrödinger Equation

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Abstract

A novel class of high-order linearly implicit energy-preserving integrating factor Runge–Kutta methods are proposed for the nonlinear Schrödinger equation. Based on the idea of the scalar auxiliary variable approach, the original equation is first reformulated into an equivalent form which satisfies a quadratic energy. The spatial derivatives of the system are then approximated with the standard Fourier pseudo-spectral method. Subsequently, we apply the extrapolation technique/prediction–correction strategy to the nonlinear terms of the semi-discretized system and a linearized energy-conserving system is obtained. A fully discrete scheme is gained by further using the integrating factor Runge–Kutta method to the resulting system. We show that, under certain circumstances for the coefficients of a Runge–Kutta method, the proposed scheme can produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution and is extremely efficient in the sense that only linear equations with constant coefficients need to be solved at every time step. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other existing structure-preserving schemes.

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Acknowledgements

The authors would like to express sincere gratitude to the referees for their insightful comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11901513, 11971481, 12071481), the National Key R&D Program of China (Grant No. 2020YFA0709800), the Natural Science Foundation of Hunan (Grant Nos. 2021JJ40655, 2021JJ20053), the Yunnan Fundamental Research Projects (Nos. 202101AT070208, 202001AT070066, 202101AS070044), the High Level Talents Research Foundation Project of Nanjing Vocational College of Information Technology (Grant No. YB20200906), and the Foundation of Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems (Grant No. 202102). Jiang and Cui are in particular grateful to Prof. Yushun Wang and Dr. Yuezheng Gong for fruitful discussions.

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Authors and Affiliations

  1. Department of Mathematics, College of Liberal Arts and Science, National University of Defense Technology, Changsha, 410073, People’s Republic of China

    Chaolong Jiang, Xu Qian & Songhe Song

  2. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, People’s Republic of China

    Chaolong Jiang

  3. Department of Basic Sciences, Nanjing Vocational College of Information Technology, Nanjing, 210023, People’s Republic of China

    Jin Cui

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  1. Chaolong Jiang
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Jiang, C., Cui, J., Qian, X. et al. High-Order Linearly Implicit Structure-Preserving Exponential Integrators for the Nonlinear Schrödinger Equation. J Sci Comput 90, 66 (2022). https://doi.org/10.1007/s10915-021-01739-x

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