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Low-regularity integrators for nonlinear Dirac equations


Authors: Katharina Schratz, Yan Wang and Xiaofei Zhao
Journal: Math. Comp. 90 (2021), 189-214
MSC (2010): Primary 35Q41, 65M12, 65M70
DOI: https://doi.org/10.1090/mcom/3557
Published electronically: August 7, 2020
MathSciNet review: 4166458
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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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Additional Information

Katharina Schratz
Affiliation: Heriot-Watt University and LJLL (UMR 7598), Sorbonne Université, UPMC, 4 place Jussieu 75005 Paris, France
MR Author ID: 990639
Email: katharina.schratz@ljll.math.upmc.fr

Yan Wang
Affiliation: School of Mathematics and Statistics, Central China Normal University, 430079 Wuhan, People’s Republic of China
Email: wang.yan@mail.ccnu.edu.cn

Xiaofei Zhao
Affiliation: School of Mathematics and Statistics; and Computational Sciences Hubei Key Laboratory, Wuhan University, 430072 Wuhan, People’s Republic of China
MR Author ID: 1045425
Email: matzhxf@whu.edu.cn

DOI: https://doi.org/10.1090/mcom/3557
Keywords: Nonlinear Dirac equation, Dirac--Poisson system, exponential-type integrator, low regularity, optimal convergence, splitting schemes.
Received by editor(s): June 22, 2019
Received by editor(s) in revised form: March 24, 2020
Published electronically: August 7, 2020
Additional Notes: The first author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 850941).
The second author was supported by the Fundamental Research Funds for the Central Universities CCNU19TD010.
The second author is the corresponding author.
The third author was partially supported by the Natural Science Foundation of Hubei Province No. 2019CFA007 and the NSFC 11901440.
Article copyright: © Copyright 2020 American Mathematical Society