An efficient and stable numerical method for the Maxwell–Dirac system

In this paper, we present an explicit, unconditionally stable and accurate numerical method for the Maxwell–Dirac system (MD) and use it to study dynamics of MD. As preparatory steps, we take the three-dimensional (3D) Maxwell–Dirac system, scale it to obtain a two-parameter model and review plane wave solution of free MD. Then we present a time-splitting spectral method (TSSP) for MD. The key point in the numerical method is based on a time-splitting discretization of the Dirac system, and to discretize nonlinear wave-type equations by pseudospectral method for spatial derivatives, and then solving the ordinary differential equations (ODEs) in phase space analytically under appropriate chosen transmission conditions between different time intervals. The method is explicit, unconditionally stable, time reversible, time transverse invariant, and of spectral-order accuracy in space and second-order accuracy in time. Moreover, it conserves the particle density exactly in discretized level and gives exact results for plane wave solution of free MD. Extensive numerical tests are presented to confirm the above properties of the numerical method. Furthermore, the tests also suggest the following meshing strategy (or ε-resolution) is admissible in the `nonrelativisticʹ limit regime (0<ε≪1): spatial mesh size h=O(ε) and time step △t=O(ε2), where the parameter ε is inversely proportional to the speed of light.