| Abstract: | 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 numerical 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 conservative 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 $\mathcal{O}(τ^2 + h^2)$ in $l^∞$-norm with time step $τ$ and mesh size $h.$ Numerical experiments
are carried out to support our theoretical conclusions. |
