Journal of Scientific Computing

, Volume 71, Issue 3, pp 1094–1134 | Cite as

Numerical Methods and Comparison for the Dirac Equation in the Nonrelativistic Limit Regime

  • Weizhu Bao
  • Yongyong CaiEmail author
  • Xiaowei Jia
  • Qinglin Tang


We analyze rigorously error estimates and compare numerically spatial/temporal resolution of various numerical methods for the discretization of the Dirac equation in the nonrelativistic limit regime, involving a small dimensionless parameter \(0<\varepsilon \ll 1\) which is inversely proportional to the speed of light. In this limit regime, the solution is highly oscillatory in time, i.e. there are propagating waves with wavelength \(O(\varepsilon ^2)\) and O(1) in time and space, respectively. We begin with several frequently used finite difference time domain (FDTD) methods and obtain rigorously their error estimates in the nonrelativistic limit regime by paying particular attention to how error bounds depend explicitly on mesh size h and time step \(\tau \) as well as the small parameter \(\varepsilon \). Based on the error bounds, in order to obtain ‘correct’ numerical solutions in the nonrelativistic limit regime, i.e. \(0<\varepsilon \ll 1\), the FDTD methods share the same \(\varepsilon \)-scalability on time step and mesh size as: \(\tau =O(\varepsilon ^3)\) and \(h=O(\sqrt{\varepsilon })\). Then we propose and analyze two numerical methods for the discretization of the Dirac equation by using the Fourier spectral discretization for spatial derivatives combined with the symmetric exponential wave integrator and time-splitting technique for temporal derivatives, respectively. Rigorous error bounds for the two numerical methods show that their \(\varepsilon \)-scalability is improved to \(\tau =O(\varepsilon ^2)\) and \(h=O(1)\) when \(0<\varepsilon \ll 1\). Extensive numerical results are reported to support our error estimates.


Dirac equation Nonrelativistic limit regime Finite difference time domain method Symmetric exponential wave integrator Time splitting Spectral method \(\varepsilon \)-Scalability 



Part of this work was done when the authors were visiting the Institute for Mathematical Sciences at the National University of Singapore in 2015.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Weizhu Bao
    • 1
  • Yongyong Cai
    • 2
    • 3
    Email author
  • Xiaowei Jia
    • 1
  • Qinglin Tang
    • 4
  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.Beijing Computational Science Research CenterBeijingPeople’s Republic of China
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA
  4. 4.Institut Elie Cartan de Lorraine, Inria Nancy-Grand EstUniversité de LorraineVandoeuvre-lès-nancy CedexFrance

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