Explicit high-order structure-preserving algorithms for the two-dimensional fractional nonlinear Schrödinger equation. Issue 5 (4th May 2022) - Fu, Yayun; Shi, Yanhua; Zhao, Yanmin
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.