A fourth-order difference scheme for the fractional nonlinear Schrödinger equation with wave operator

ABSTRACT

In this paper, an efficient semi-implicit difference scheme for solving the fractional nonlinear Schrödinger equation with wave operator are proposed and analyzed. The semi-implicit scheme involves three-time levels, is unconditionally stable and fourth-order accurate in space and second-order accurate in time. Furthermore, the unique solvability, unconditional stability and convergence of the method in the $L^{\mathrm{\infty }}$-norm are proved rigorously by the energy method. Finally, numerical experiments are presented to confirm our theoretical results.

Keywords:

• Nonlinear Schrödinger equation
• fractional Laplacian
• wave operator
• unconditional stability
• semi-implicit difference scheme

2010 Mathematics Subject Classification:

• 65M06

Acknowledgments

The authors would like to thank the two anonymous referees for valuable suggestions, which helped to improve the quality of the paper. Kejia Pan was supported by Science Challenge Project (No. TZ2016002), the National Natural Science Foundation of China (No. 41874086), the Excellent Youth Foundation of Hunan Province of China (No. 2018JJ1042). Dongdong He was supported by the president's fund-research start-up fund from the Chinese University of Hong Kong, Shenzhen (No. PF01000857).

Disclosure statement

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

Additional information

Funding

Kejia Pan was supported by Science Challenge Project [grant number TZ2016002], and the National Natural Science Foundation of China [grant number 41874086], the Excellent Youth Foundation of Hunan Province of China [grant number 2018JJ1042]. Dongdong He was supported by the president's fund-research start-up fund from the Chinese University of Hong Kong, Shenzhen [grant number PF01000857].