Low-regularity integrators for nonlinear Dirac equations

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 an 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, a 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 a 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.