Low-regularity integrators for nonlinear Dirac equations

Schratz, Katharina; Wang, Yan; Zhao, Xiaofei

doi:10.1090/mcom/3557

Low-regularity integrators for nonlinear Dirac equations
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by Katharina Schratz, Yan Wang and Xiaofei Zhao;

Math. Comp. 90 (2021), 189-214

DOI: https://doi.org/10.1090/mcom/3557

Published electronically: August 7, 2020

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Abstract:

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.

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