Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System

In this paper, we derive and analyze a conservative Crank-Nicolson-type finite difference scheme for the Klein-Gordon-Dirac (KGD) system. Differing from the derivation of the existing numerical methods given in literature where the nu-merical schemes are proposed by directly discretizing the KGD system, we translate the KGD equations into an equivalent system by introducing an auxiliary function, then derive a nonlinear Crank-Nicolson-type finite difference scheme for solving the equivalent system. The scheme perfectly inherits the mass and energy conserva-tive properties possessed by the KGD, while the energy preserved by the existing conservative numerical schemes expressed by two-level's solution at each time step. By using energy method together with the 'cut-off' function technique, we establish the optimal error estimate of the numerical solution, and the convergence rate is O(tau 2 + h2) in l infinity-norm with time step tau and mesh size h. Numerical experiments are carried out to support our theoretical conclusions.