期刊论文

Schratz, Katharina

英文

MATHEMATICS OF COMPUTATION

0025-5718

2021

90

327

189-214

10.1090/mcom/3557

Received by the editor June 22, 2019, and, in revised form, March 24, 2020. 2010 Mathematics Subject Classification. Primary 35Q41, 65M12, 65M70. Key words and phrases. Nonlinear Dirac equation, Dirac–Poisson system, exponential-type integrator, low regularity, optimal convergence, splitting schemes. 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.

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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 ...