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Optimal Resolution Methods for the Klein–Gordon–Dirac System in the Nonrelativistic Limit Regime

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Abstract

We propose and compare numerically spatial/temporal resolution of various efficient numerical methods for solving the Klein–Gordon–Dirac system (KGD) in the nonrelativistic limit regime. The KGD system involves a small dimensionless parameter \(0<\varepsilon \ll 1\) in this limit regime and admits rapid oscillations in time as \(\varepsilon \rightarrow 0^+\). By adopting the Fourier spectral discretization for spatial derivatives followed with the time-splitting or exponential wave integrators based on some efficient quadrature rules in phase field, we propose four different numerical discretizations for the KGD system. The discretizations are all fully explicit and valid in one, two and three dimensions. Extensive numerical results demonstrate that these discretizations provide optimal numerical resolutions for the KGD system, i.e., under the mesh strategies \(\tau =O(\varepsilon ^2)\) and \(h=O(1)\) with time step \(\tau \) and mesh size h in terms of \(\varepsilon \), they all perform well with uniform spectral accuracy in space and second-order accuracy in time. In addition, the \(\varepsilon \)-scalability of the best method is improved as \(\tau =O(\varepsilon )\), which is much superior than that of the finite difference methods. For applications, we profile the dynamics of the KGD system in 2D with a honeycomb lattice potential, which depend greatly on the singular perturbation \(\varepsilon \) and the weak/strong interaction.

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Acknowledgements

The authors would like to thank Professor Weizhu Bao and Professor Yongyong Cai for their valuable suggestions and comments. Part of this work was done when the authors were visiting the Department of Mathematics at the National University of Singapore in 2017.

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Authors and Affiliations

  1. Institute of Mathematics, College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

    Wenfan Yi

  2. Beijing Computational Science Research Center, No. 10 West Dongbeiwang Road, Haidian District, Beijing, 100193, China

    Wenfan Yi

  3. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Xinran Ruan

  4. Laboratoire J.-L. Lions, Université Pierre et Marie Curie, 75252, Paris cedex 05, France

    Xinran Ruan

  5. Department of Mathematics, University of Innsbruck, 6020, Innsbruck, Austria

    Chunmei Su

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  1. Wenfan Yi
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Correspondence to Xinran Ruan.

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This work was supported by the Fundamental Research Funds for the Central Universities 531107051208 and the NSFC Grant 11601148.

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Yi, W., Ruan, X. & Su, C. Optimal Resolution Methods for the Klein–Gordon–Dirac System in the Nonrelativistic Limit Regime. J Sci Comput 79, 1907–1935 (2019). https://doi.org/10.1007/s10915-019-00919-0

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