  In this work, we consider the numerical integration of the nonlinear Dirac
equation and the Dirac-Poisson system (NDEs) under rough initial data. We
propose a ultra low-regularity integrator (ULI) for solving the NDEs which
enables optimal first-order time convergence in $H^r$ for solutions in $H^{r}$,
i.e., without requiring any additional regularity on the solution. In contrast
to classical methods, ULI overcomes the numerical loss of derivatives and is
therefore more efficient and accurate for approximating low regular solutions.
Convergence theorems and the extension of ULI to second order are established.
Numerical experiments confirm the theoretical results and underline the
favourable error behaviour of the new method at low regularity compared to
classical integration schemes.
