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Error Bounds of the Finite Difference Time Domain Methods for the Dirac Equation in the Semiclassical Regime

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Abstract

We study rigorously the error bounds of four frequently-used finite difference time domain (FDTD) methods for the Dirac equation in the semiclassical regime, involving a small dimensionless parameter \(0<\varepsilon \le 1\) representing the scaled Planck constant. In this regime, there are highly oscillatory propagating waves with wavelength \(O(\varepsilon )\) in both time and space of the solution. We apply the leap-frog, two semi-implicit, and the Crank–Nicolson finite difference methods to numerically solve the equation, and establish rigorously their error estimates. We prove that these methods share the same error bounds, which are explicitly related to time step size \(\tau \), mesh size h, as well as the small parameter \(\varepsilon \). Furthermore, we find out the dependence of the observables, i.e. the total probability density and the current density on the parameters \(\tau \), h and \(\varepsilon \). Based on the error bounds, in the semiclassical regime, i.e. \(0<\varepsilon \ll 1\), to obtain ‘correct’ numerical solutions and related observables, the \(\varepsilon \)-scalabilities \(\tau = O(\varepsilon ^{3/2})\) and \(h = O(\varepsilon ^{3/2})\) are required for all these FDTD methods. Numerical results are carried out to support our error estimates.

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Authors and Affiliations

  1. Beijing Computational Science Research Center, Beijing, 100193, People’s Republic of China

    Ying Ma

  2. NUS Graduate School for Integrative Sciences and Engineering (NGS), National University of Singapore, Singapore, 119077, Singapore

    Jia Yin

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  1. Ying Ma
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  2. Jia Yin
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This work was supported by the Ministry of Education of Singapore Grant R-146-000-247-114 (J. Yin)

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Ma, Y., Yin, J. Error Bounds of the Finite Difference Time Domain Methods for the Dirac Equation in the Semiclassical Regime. J Sci Comput 81, 1801–1822 (2019). https://doi.org/10.1007/s10915-019-01063-5

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