An energy-preserving scheme for coupled fractional Gross-Pitaevskii equations based on energy discretization

We present an efficient structure-preserving numerical scheme for the coupled fractional Gross-Pitaevskii (CFGP) equations. The basic idea is from the discrete variational derivative method to construct energy-stable schemes for partial differential systems. We here extend the discretization of the local energy at the traditional grid points (x_j,t_n) to the midpoints (x_j,t_n+1/2). The local energy equation is then discretized in space using the Fourier pseudo-spectral method. A new discrete variational derivative system is established at the order of O(M^α/2−r + τ²) with the spatial grid sizes M and time step τ. Furthermore, the conservation properties, boundedness, existence, and convergence of the numerical scheme are rigorously proven. The Richardson extrapolation method is then employed to improve the temporal accuracy from second to fourth order, and mass conservation of the extrapolated scheme is theoretically established. Numerical experiments are provided to verify the theoretical analysis results, as well as the conservation properties before and after applying the Richardson extrapolation.