A linearly implicit conservative scheme for the fractional nonlinear Schrödinger equation with wave operator

Abstract

The present work is mainly devoted to studying the fractional nonlinear Schrödinger equation with wave operator. We first derive two conserved quantities of the equation, and then develop a three-level linearly implicit difference scheme. This scheme is shown to be conserves the discrete version of conserved quantities. Using energy method, we prove that the difference scheme is unconditionally stable, and the difference solution converges to the exact one with second order accuracy in both the space and time dimensions. Numerical experiments are performed to support our theoretical analysis and demonstrate the accuracy, discrete conservation laws and effectiveness for long-time simulation.

Keywords:

• fractional Schrödinger equation
• fractional centred difference
• stability
• convergence
• conservation

2010 AMS Subject Classifications:

• 35Q41
• 65M06
• 65M12

Acknowledgements

The authors are very grateful to the referees for their constructive comments and valuable suggestions, which greatly improved the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

This work was supported by National Natural Science Foundation of China under Grant [11171125], [91130003].