Uniform error bounds of exponential wave integrator methods for the long-time dynamics of the Dirac equation with small potentials

Two exponential wave integrator Fourier pseudospectral (EWI-FP) methods are presented and analyzed for the long-time dynamics of Dirac equation with small potentials characterized by ??(0,1] a dimensionless parameter. Based on (symmetric) temporal derivatives in phase space followed applying discretization spatial derivatives, EWI-FP explicit spectral accuracy second-order time any fixed ?=?0. Uniform error bounds rigorously carried out at O(hm0+?2) up to O(1/?) mesh size h, step ? m0 an integer depending regularity solution. Extensive numerical results reported confirm our comparisons two shown. Finally, 2D validate schemes.