An efficient and accurate Fourier pseudo-spectral method for the nonlinear Schrödinger equation with wave operator

Abstract

In this study, we design and analyse an efficient and accurate Fourier pseudo-spectral method for solving the nonlinear Schrödinger equation with wave operator (NLSW). In this method, a modified leap-frog finite difference method is adopted for time discretization and a Fourier pseudo-spectral method is employed for spatial discretization. A new type of discrete energy functional is defined in a recursive way. It is then proved that the proposed method preserves the total energy in the discrete sense. Thanks to the FFT algorithm, the proposed method can be explicitly computed in the practical computation. Besides the standard energy method, the key tools used in our numerical analysis are a mathematical induction argument and a lifting technique. Without any restriction on the grid ratio, we prove that the proposed method is of spectral accuracy in space and second-order accuracy in time. Numerical results are reported to verify the error estimate and energy conservation of the proposed method.

Keywords:

• Schrödinger equation with wave operator
• Fourier pseudo-spectral method
• unconditional convergence
• optimal error estimate
• energy conservation

2010 Mathematics Subject Classifications:

• 65M06
• 65M12
• 65M15
• 65M70

Acknowledgments

Tingchun Wang's work is supported by the National Natural Science Foundation of China (Grant No. 11571181) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20171454); Jialing Wang's work is supported by the National Natural Science Foundation of China (Grant No.11801277) and the Startup Foundation for Introducing Talent of NUIST (Grant No. 2017r090).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Tingchun Wang's work is supported by the National Natural Science Foundation of China (Grant No. 11571181) and the Natural Science Foundation of Jiangsu Province (Grant No. BK20171454); Jialing Wang's work is supported by the National Natural Science Foundation of China (Grant No. 11801277) and the Startup Foundation for Introducing Talent of NUIST (Grant No. 2017r090).