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Two exponential wave integrator Fourier pseudospectral (EWI-FP) methods arepresented and analyzed for the long-time dynamics of the Dirac equation withsmall potentials characterized by $\varepsilon \in (0, 1]$ a dimensionlessparameter. Based on the (symmetric) exponential wave integrator for temporalderivatives in phase space followed by applying the Fourier pseudospectraldiscretization for spatial derivatives, the EWI-FP methods are explicit and ofspectral accuracy in space and second-order accuracy in time for any fixed$\varepsilon = \varepsilon_0$. Uniform error bounds are rigorously carried outat $O(h^{m_0}+\tau^2)$ up to the time at $O(1/\varepsilon)$ with the mesh size$h$, time step $\tau$ and $m_0$ an integer depending on the regularity of thesolution. Extensive numerical results are reported to confirm our error boundsand comparisons of two methods are shown. Finally, dynamics of the Diracequation in 2D are presented to validate the numerical schemes.

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