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We compare the long-time error bounds and spatial resolution of finitedifference methods with different spatial discretizations for the Diracequation with small electromagnetic potentials characterized by $\varepsilon\in (0, 1]$ a dimensionless parameter. We begin with the simple and widely usedfinite difference time domain (FDTD) methods, and establish rigorous errorbounds of them, which are valid up to the time at $O(1/\varepsilon)$. In theerror estimates, we pay particular attention to how the errors dependexplicitly on the mesh size $h$ and time step $\tau$ as well as the smallparameter $\varepsilon$. Based on the results, in order to obtain "correct"numerical solutions up to the time at $O(1/\varepsilon)$, the$\varepsilon$-scalability (or meshing strategy requirement) of the FDTD methodsshould be taken as $h = O(\varepsilon^{1/2})$ and $\tau =O(\varepsilon^{1/2})$. To improve the spatial resolution capacity, we apply theFourier spectral method to discretize the Dirac equation in space. Error boundsof the resulting finite difference Fourier pseudospectral (FDFP) methods showthat they exhibit uniform spatial errors in the long-time regime, which areoptimal in space as suggested by the Shannon's sampling theorem. Extensivenumerical results are reported to confirm the error bounds and demonstrate thatthey are sharp.

Spatial resolution of different discretizations over long-time for the Dirac equation with small potentials