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Convergence analysis of a symmetric exponential integrator Fourier pseudo-spectral scheme for the Klein–Gordon–Dirac equation

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  • Li, Jiyong

Abstract

Recently, an exponential integrator Fourier pseudo-spectral (EIFP) scheme for the Klein–Gordon–Dirac (KGD) equation in the nonrelativistic limit regime has been proposed (Yi et al., 2019). The scheme is fully explicit and numerical experiments show that it is very efficient due to the fast Fourier transform (FFT). However, the authors did not give a strict convergence analysis and error estimate for the scheme. In addition, the scheme did not satisfy time symmetry which is an important characteristic of the exact solution. In this paper, by setting two-level format for Klein–Gordon part and three-level format for Dirac part, respectively, we proposed a new EIFP scheme for the KGD equation with periodic boundary conditions. The new scheme is time symmetric and fully explicit. By using the standard energy method and the mathematical induction, we make a rigorously convergence analysis and establish error estimates without any CFL condition restrictions on the grid ratio. The convergence rate of the proposed scheme is proved to be at the second-order in time and spectral-order in space, respectively, in a generic Hm-norm. The numerical experiments are carried out to confirm our theoretical analysis. Because that our error estimates are given under the general Hm-norm, the conclusion can easily be extended to two- and three-dimensional problems without the stability (or CFL) condition under sufficient regular conditions.

Suggested Citation

  • Li, Jiyong, 2021. "Convergence analysis of a symmetric exponential integrator Fourier pseudo-spectral scheme for the Klein–Gordon–Dirac equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 691-713.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:691-713
    DOI: 10.1016/j.matcom.2021.06.007
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    References listed on IDEAS

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    1. Li, Jiyong, 2017. "A family of improved Falkner-type methods for oscillatory systems," Applied Mathematics and Computation, Elsevier, vol. 293(C), pages 345-357.
    2. Ömer Oruç & Alaattin Esen & Fatih Bulut, 2016. "A Haar wavelet collocation method for coupled nonlinear Schrödinger–KdV equations," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 27(09), pages 1-16, September.
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    Cited by:

    1. Li, Jiyong, 2023. "Optimal error estimates of a time-splitting Fourier pseudo-spectral scheme for the Klein–Gordon–Dirac equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 208(C), pages 398-423.
    2. Wang, Xianfen & Li, Jiyong, 2023. "Convergence analysis of two conservative finite difference fourier pseudo-spectral schemes for klein-Gordon-Dirac system," Applied Mathematics and Computation, Elsevier, vol. 439(C).

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