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Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

Authors: Weizhu Bao, Yongyong Cai and Jia Yin
Journal: Math. Comp. 89 (2020), 2141-2173
MSC (2010): Primary 35Q41, 65M70, 65N35, 81Q05
Published electronically: April 23, 2020
MathSciNet review: 4109563
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Abstract: We establish error bounds of the Lie-Trotter splitting ($ S_1$) and Strang splitting ($ S_2$) for the Dirac equation in the nonrelativistic regime in the absence of external magnetic potentials, with a small parameter $ 0<\varepsilon \leq 1$ inversely proportional to the speed of light. In this regime, the solution propagates waves with $ O(\varepsilon ^2)$ wavelength in time. Surprisingly, we find out that the splitting methods exhibit super-resolution, i.e., the methods can capture the solutions accurately even if the time step size $ \tau $ is independent of $ \varepsilon $, while the wavelength in time is at $ O(\varepsilon ^2)$. $ S_1$ shows $ 1/2$ order convergence uniformly with respect to $ \varepsilon $, by establishing that there are two independent error bounds $ \tau + \varepsilon $ and $ \tau + \tau /\varepsilon $. Moreover, if $ \tau $ is nonresonant, i.e., $ \tau $ is away from a certain region determined by $ \varepsilon $, $ S_1$ would yield an improved uniform first order $ O(\tau )$ error bound. In addition, we show $ S_2$ is uniformly convergent with $ 1/2$ order rate for general time step size $ \tau $ and uniformly convergent with $ 3/2$ order rate for nonresonant time step size. Finally, numerical examples are reported to validate our findings.

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Additional Information

Weizhu Bao
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119076

Yongyong Cai
Affiliation: School of Mathematical Sciences, Beijing Normal University, 100875, People’s Republic of China; and Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China

Jia Yin
Affiliation: NUS Graduate School for Integrative Sciences and Engineering (NGS), National University of Singapore, Singapore 117456

Keywords: Dirac equation, super-resolution, nonrelativistic regime, time-splitting, uniform error bound
Received by editor(s): February 21, 2019
Received by editor(s) in revised form: December 4, 2019, and January 18, 2020
Published electronically: April 23, 2020
Additional Notes: This work was partially done when the first author was visiting the Courant Institute for Mathematical Sciences in 2018. Part of this work was done when the authors visited the Institute for Mathematical Sciences, National University of Singapore, in 2019.
The first and third authors acknowledge support from the Ministry of Education of Singapore grant R-146-000-247-114
The second author acknowledges support from NSFC grant No. 11771036 and 91630204.
Article copyright: © Copyright 2020 American Mathematical Society