TY - JOUR T1 - Error Estimate of a New Conservative Finite Difference Scheme for the Klein-Gordon-Dirac System AU - Bian , Shasha AU - Cheng , Yue AU - Guo , Boling AU - Wang , Tingchun JO - Numerical Mathematics: Theory, Methods and Applications VL - 1 SP - 140 EP - 164 PY - 2023 DA - 2023/01 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2022-0049 UR - https://global-sci.org/intro/article_detail/nmtma/21346.html KW - Klein-Gordon-Dirac equation, nonlinear finite difference scheme, conservation, error analysis. AB -

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.