Analysis of a Fourier pseudo-spectral conservative scheme for the Klein-Gordon-Schrodinger equation

In this paper, we focus on constructing and analysing a new Fourier pseudo-spectral conservative scheme for the Klein-Gordon-Schrodinger (KGS) equation. After rewriting the KGS equation as an infinite-dimensional Hamiltonian system, we use a Fourier pseudo-spectral method to discrete the system in space to obtain a semi-discrete system, which can be cast into a canonical finite-dimensional Hamiltonian form. Then, an energy-preserving and charge-preserving scheme is constructed by using the symmetric discrete gradient method. Based on the discrete conservation laws and the equivalence of the semi-norm between the Fourier pseudo-spectral method and the finite difference method, the pseudo-spectral solution of the proposed scheme is proved to be bounded in the discrete L-infinity norm. The proposed scheme is shown to be convergent with the convergence order of O(J(-r) + tau(2)) in the discrete L-2 norm afterwards, where J is the number of nodes and tau is the time step size. Numerical experiments are conducted to verify the theoretical analysis.