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A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with structure-preservation for long-time dynamics of the Dirac equation with small potentials

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Abstract

For the Dirac equation with potentials characterized by a small parameter ε ∈ (0,1], the numerical methods for long-time dynamics have received more and more attention. Recently, two exponential wave integrator Fourier pseudo-spectral (EWIFP) methods for the Dirac equation have been proposed (Feng et al., Appl. Numer. Math. 172, 50–66, 2022) which are uniformly accurate about ε and perform well over the classical methods. However, the EWIFP methods cannot preserve the mass and energy, which are important structural features of the Dirac equation from the perspective of geometric numerical integration. In addition, the EWIFP methods are not time symmetric or only are conditionally stable under specific stability condition which implies CFL condition restrictions on the grid ratio. In this work, we propose a structure-preserving EWIFP (SPEWIFP) method. The proposed method is proved to be time symmetric, stable only under the condition \(\tau \lesssim 1\), and preserves the discrete energy and modified mass. Without any CFL condition restrictions on the grid ratio, we carry out a rigourously error analysis and give uniform error bounds of the method at \(O(h^{m_{0}} + \varepsilon ^{1-\beta } \tau ^{2})\) up to the time at O(1/εβ) with β ∈ [0,1], mesh size h, time step τ, and an integer m0 determined by the regularity conditions. In general, the Dirac equation with small potentials can be converted to an oscillatory Dirac equation with wavelength at O(εβ) in time which includes the case of simultaneous massless and nonrelativistic regime. It is easy to extend the error bounds and structure-preservation properties to the oscillatory Dirac equation. Numerical experiments support our error bounds and structure-preservation properties.

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Acknowledgements

The authors are grateful to the anonymous reviewers for their valuable suggestions, which help improve this paper significantly.

Funding

The research was supported in part by the Natural Science Foundation of Hebei Province (No. A2021205036) and Education Department of Hebei Province Fund (No. QN2019053).

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Authors and Affiliations

  1. College of Mathematics Science, Hebei Normal University, Shijiazhuang, 050024, People’s Republic of China

    Jiyong Li & Liqing Zhu

  2. Hebei International Joint Research Center for Mathematics and Interdisciplinary Science, Shijiazhuang, 050024, People’s Republic of China

    Jiyong Li

  3. Hebei Key Laboratory of Computational Mathematics and Applications, Shijiazhuang, 050024, People’s Republic of China

    Jiyong Li

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  1. Jiyong Li
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Li, J., Zhu, L. A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with structure-preservation for long-time dynamics of the Dirac equation with small potentials. Numer Algor 92, 1367–1401 (2023). https://doi.org/10.1007/s11075-022-01345-4

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