Some energy-preserving relaxation-type schemes for two-dimensional space fractional nonlinear Schrödinger equations

In this paper, we develop some energy-preserving relaxation-type schemes to solve two-dimensional space fractional nonlinear Schrödinger equations with periodic boundary conditions. First, we change the original system into an equivalent relaxation form by introducing some new variables, which transforms the energy conservation law of the original system into quadratic invariants, and satisfies the mass conservation law of the original system. Then an implicit relaxation scheme is applied to deal with the time derivative, and the resulting semi discrete system can exactly preserve the mass and energy conservation laws. However, the obtained semi discrete system is nonlinear. Next a linear implicit relaxation scheme is directly used for the modified system to arrive at a semi discrete scheme, and the conservation of the semi discrete system is analyzed. Second, the resulting semi discrete systems are discretized by the Fourier spectral method with periodic boundary conditions, and the efficient iterative algorithms of the fully-discrete systems are given. Finally, numerical experiments of some space fractional nonlinear Schrödinger equations are given to verify the correctness of the theoretical results.