Efficient energy-preserving finite difference schemes for the Klein-Gordon-Schrödinger equations

In this study, we construct three efficient and conservative high-order accurate finite difference schemes for solving the Klein-Gordon-Schrodinger equations with homogeneous Dirichlet boundary conditions. The spatial discretization is carried out by a novel fourth-order accurate central difference scheme in which the fast discrete sine transform can be utilized for efficient implementation. The second-order conservative Crank-Nicolson scheme is considered in the temporal direction. Then a priori estimate, conservation laws, and convergence of the first scheme in two-dimensional space are discussed. A linearized iteration based on the fast discrete sine transform technique is derived to solve the nonlinear system effectively. Because the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. For comparative purposes, two other schemes are constructed based on improved scalar auxiliary variable approaches by converting the Klein-Gordon-Schrodinger equations into an equivalent new system which involves solving linear systems with constant coefficients at each time step. Moreover, we need to point out that the proposed schemes are decoupled, which makes them appropriate for parallel computation to significantly reduce the computing time. Finally, numerical experiments are presented to validate the correctness of theoretical results and demonstrate the excellent performance in long-time conservation of the schemes.