Abstract
In this paper, we study a fast explicit operator splitting method for space fractional nonlinear Schrödinger equation in one (1D), two (2D) and three dimensions (3D) with periodic boundary conditions. The equation is split into linear and nonlinear parts: the linear part is solved by the Fourier spectral method, which is based on the exact solution and thus has no stability restriction on the time-step size; the nonlinear subequation is then solved analytically due to the availability of a closed-form solution. The rigorous analysis of the discrete mass conservation principle and the convergence rate of the proposed algorithm are proved. The theoretical results show the proposed method is \(L^2\) unconditionally stable, second order accurate in time, whereas the spatial accuracy depends on the regularity of the solution. Numerical experiments for 1D, 2D and 3D cases demonstrate the efficiency of the proposed method.
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Acknowledgements
The authors would like to gratefully thank Professor Wei Wei from Northwest University for the helpful discussions on the regularity of SFNLS equation in this work. The authors also thank the anonymous referees for their valuable comments and suggestions, which were very helpful in improving the quality of the paper.
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This work is in part supported by the National Natural Science Foundation of China (Nos. 11701196, 11871057, 11501447, 11701197, 11701081, 1186010285), the Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University (No. ZQN-YX502), the Fundamental Research Funds for the Central Universities (Nos. ZQN-702, 2242019K40111), Natural Science Youth Foundation of Jiangsu Province (No. BK20160660), the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (No. BM2017002), Key Project of Natural Science Foundation of China (No. 61833005).
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Zhai, S., Wang, D., Weng, Z. et al. Error Analysis and Numerical Simulations of Strang Splitting Method for Space Fractional Nonlinear Schrödinger Equation. J Sci Comput 81, 965–989 (2019). https://doi.org/10.1007/s10915-019-01050-w
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DOI: https://doi.org/10.1007/s10915-019-01050-w
Keywords
- Fractional nonlinear Schrödinger equation
- Operator splitting method
- Spectral method
- Mass conservation
- Error estimate