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Improved Error Bounds of the Strang Splitting Method for the Highly Oscillatory Fractional Nonlinear Schrödinger Equation

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Abstract

We begin with rigorous error estimates of the Strang splitting method for the highly oscillatory fractional nonlinear equation involving a small parameter \(\varepsilon \in (0, 1]\), which propagates waves with wavelength at \(O(\varepsilon ^2)\) in time. In view of the inherent oscillatory nature, the \(\varepsilon \)-scalability of the Strang splitting method is optimal as suggested by the Shannon’s sampling theorem. Surprisingly, we find out that the Strang splitting method would yield an improved error bound for the one dimensional (1D) fractional nonlinear Schrödinger equation (NLSE) provided that the time step \(\tau \) is chosen as an integer fraction of the period of the principal linear part. Finally, numerical examples are reported to validate our error estimates and various applications are shown to illustrate the difference between the fractional NLSE and classical NLSE as well as the capability of the Strang splitting method.

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Acknowledgements

The author would like to specially thank Professor Weizhu Bao for his valuable suggestions and comments. This work was supported by the Ministry of Education of Singapore Grant R-146-000-290-114.

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  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Yue Feng

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Feng, Y. Improved Error Bounds of the Strang Splitting Method for the Highly Oscillatory Fractional Nonlinear Schrödinger Equation. J Sci Comput 88, 48 (2021). https://doi.org/10.1007/s10915-021-01558-0

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  • DOI: https://doi.org/10.1007/s10915-021-01558-0

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