A numerical method with particle conservation for the Maxwell–Dirac system

A numerical method is presented for solving the Maxwell–Dirac systems. The Maxwell equations with particle and current densities as the source terms are discretized explicitly. To guarantee the particle conservation, the Dirac equations coupled electromagnetic potentials are discretized by the time-splitting method and implicit finite difference. These numerical schemes are conservative in particle density and have second-order accuracy in time and space. One-dimensional numerical results are given to validate the accuracy and the conservation and three-dimensional examples are presented to describe dynamical behaviors of the Maxwell–Dirac system with several external potentials.