Error Estimates of Crank-Nicolson Finite Difference Scheme for Klein-Gordon-Schrödinger Equations with Nonlinear Damping

The purpose of this paper is to derive a Crank-Nicolson finite difference scheme for the Klein-Gordon-Schrödinger (KGS) equations with nonlinear loss. Under appropriate regularity assumptions, we establish the existence and uniqueness of the fully discrete solution by analyzing the boundedness of discrete mass and energy. Furthermore, we derive a priori error estimates in the discrete $$L^2$$ and $$H_0^1$$ norms, showing that the scheme attains second-order convergence in both time and space. Numerical experiments are conducted to validate the theoretical analysis and demonstrate the accuracy and stability of the proposed scheme.