In the paper, we study structure-preserving scheme to solve general fractional Klein-Gordon-Schrödinger equations, including one dimension case and two dimension case. First, the high central difference scheme and Crank-Nicolson scheme are used to one dimension fractional Klein-Gordon-Schrödinger equations. We show that the arising scheme is uniquely solvable, and approximate solutions converge to the exact solution at the rate $O({\tau }^2+h^4)$. Moreover, we prove that the resulting scheme can preserve the mass and energy conservation laws. Second, we show Crank-Nicolson scheme for two dimension fractional Klein-Gordon-Schrödinger equations, and the proposed scheme preserves the mass and energy conservation laws in discrete formulations. However, the obtained discrete system is nonlinear system. Then, we show a equivalent form of fractional Klein-Gordon-Schrödinger equations by introducing some new auxiliary variables. The new system is discretized by the high central difference scheme and scalar auxiliary variable scheme, and a linear discrete system is obtained, which can preserve the energy conservation law. Finally, the numerical experiments including one dimension and two dimension fractional Klein-Gordon-Schrödinger systems are given to verify the correctness of theoretical results.

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.