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.

中文翻译：

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非线性狄拉克方程的低正则积分器

在这项工作中，我们考虑了在粗略初始数据下非线性狄拉克方程和狄拉克-泊松系统 (NDE) 的数值积分。我们提出了一种用于解决 NDE 的超低正则积分器 (ULI)，它可以在 $H^r$ 中实现最优的一阶时间收敛，即 $H^{r}$ 中的解决方案，即不需要对解决方案进行任何额外的正则性. 与经典方法相比，ULI 克服了导数的数值损失，因此对于近似低正则解更有效和准确。建立收敛定理和 ULI 到二阶的扩展。数值实验证实了理论结果，并强调了与经典积分方案相比，新方法在低规律性下的有利误差行为。