Optimal error estimates of finite difference time domain methods for the Klein–Gordon–Dirac system

Abstract We propose and analyze finite difference methods for solving the Klein–Gordon–Dirac (KGD) system. Due to the nonlinear coupling between the complex Dirac ‘wave function’ and the real Klein–Gordon field, it is a great challenge to design and analyze numerical methods for KGD. To overcome the difficulty induced by the nonlinearity, four implicit/semi-implicit/explicit finite difference time domain (FDTD) methods are presented, which are time symmetric or time reversible. By rigorous error estimates, the FDTD methods converge with second-order accuracy in both spatial and temporal discretizations, and numerical results in one dimension are reported to support our conclusion. The error analysis relies on the energy method, the special nonlinear structure in KGD and the mathematical induction. Thanks to tensor grids and discrete Sobolev inequalities, our approach and convergence results are valid in higher dimensions under minor modifications.