Explicit high-order structure-preserving algorithms for the two-dimensional fractional nonlinear Schrödinger equation

Abstract

The paper aims to construct a class of high-order explicit conservative schemes for the space fractional nonlinear Schrödinger equation by combing the invariant energy quadratization method and Runge-Kutta method. We first derive the Hamiltonian formulation of the equation, and obtain a new equivalent system via introducing a scalar variable. Then, we propose a semi-discrete conservative system by using the Fourier pseudo-spectral method to approximate the equivalent system in space. Further applying the fourth-order modified Runge-Kutta method to the semi-discrete system gives two classes of schemes for the equation. One scheme preserves the energy while the other scheme conserves the mass. Numerical experiments are provided to demonstrate the conservative properties, convergence orders and long time stability of the proposed schemes.

Keywords:

• Fractional nonlinear Schrödinger equation
• structure-preserving algorithms
• invariant energy quadratization
• Runge-Kutta method
• explicit conservative schemes

AMS subject classifications:

• 35R11
• 65M70

Acknowledgments

This work is supported by the National Key Research and Development Project of China (Grant No. 2017YFC0601505, 2018YFC1504205), the National Natural Science Foundation of China (Grant No. 11771213, 11971416, 11971242), the Major Projects of Natural Sciences of University in Jiangsu Province of China (Grant No.18KJA110003), and the Program for Scientific and Technological Innovation Talents in Universities of Henan Province (No. 19HASTIT025).

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Key Program of Xuchang University (2021ZD012), the National Natural Science Foundation of China (Grant No. 11771213, 11971416, 11971242), the Major Projects of Natural Sciences of University in Jiangsu Province of China (Grant No. 18KJA110003), and the Program for Scientific and Technological Innovation Talents in Universities of Henan Province (No. 19HASTIT025).