Unconditionally stable explicit exponential methods for the Klein–Gordon–Schrödinger equations

In this paper, we present a framework to derive unconditionally stable explicit exponential methods for the coupled Klein–Gordon–Schrödinger (KGS) equations. The approach is based on the Hamiltonian or operator splitting. By splitting the KGS equations into three independently linear equations and solving these equations exactly with exponential methods after suitable spatial discretization, two kinds of explicit exponential methods are obtained, which could be of any order accuracy in time. It is proved that the proposed methods are time-symmetric, unconditionally stable, and mass-preserving. In particular, the derived Hamiltonian-splitting methods are symplectic and thus nearly preserve the energy. The convergence of the second-order (in time) methods is also proved. Moreover, we present a fast implementation with the Fast Fourier Transform (FFT) technique once periodic boundary conditions are prescribed for the KGS equations. Finally, 1D and 2D KGS equations are tested with the second-order and fourth-order (in time) methods. Numerical results demonstrate the high efficiency, unconditional stability with the independence of the mesh ratio, good energy and mass conservation, and applicability of large time stepsizes of the methods proposed in this paper.