A dissipative finite difference Fourier pseudo-spectral method for the Klein-Gordon-Schrödinger equations with damping mechanism.

We develop a semi-linearized, decoupled time-stepping method for solving the Klein-Gordon-Schrödinger equations with damping mechanism. The finite difference approximation in time and Fourier pseudo-spectral discretization in space provide an elegant platform to deal with the physical properties of the original model. We prove that the proposed numerical algorithm preserves the discrete invariant or dissipative properties of system exactly depending on the choices of the damping parameter values. We establish the maximum norm error estimates by virtue of the norm-equivalence between finite difference method and Fourier pseudo-spectral method, the discrete versions of projection and interpolation estimations, and mathematical induction argument. Ample numerical results are presented to show the effectiveness of our numerical method and to confirm our theoretical analysis.