A fourth-order compact time-splitting Fourier pseudospectral method for the Dirac equation

  • Weizhu Bao
  • Jia YinEmail author


We propose a new fourth-order compact time-splitting (\(S_\mathrm{4c}\)) Fourier pseudospectral method for the Dirac equation by splitting the Dirac equation into two parts together with using the double commutator between them to integrate the Dirac equation at each time interval. The method is explicit, fourth-order in time and spectral order in space. It is unconditionally stable and conserves the total probability in the discretized level. It is called a compact time-splitting method since, at each time step, the number of substeps in \(S_\mathrm{4c}\) is much less than those of the standard fourth-order splitting method and the fourth-order partitioned Runge–Kutta splitting method. Another advantage of \(S_\mathrm{4c}\) is that it avoids to use negative time steps in integrating subproblems at each time interval. Comparison between \(S_\mathrm{4c}\) and many other existing time-splitting methods for the Dirac equation is carried out in terms of accuracy and efficiency as well as longtime behavior. Numerical results demonstrate the advantage in terms of efficiency and accuracy of the proposed \(S_\mathrm{4c}\). Finally, we report the spatial/temporal resolutions of \(S_\mathrm{4c}\) for the Dirac equation in different parameter regimes including the nonrelativistic limit regime, the semiclassical limit regime, and the simultaneously nonrelativistic and massless limit regime.


Dirac equation Fourth-order compact time-splitting Double commutator Probability conservation Nonrelativistic limit regime Semiclassical limit regime 



This work was partially supported by the Ministry of Education of Singapore Grant R-146-000-223-112 (MOE2015-T2-2-146).


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Authors and Affiliations

  1. 1.Department of MathematicsNational University of SingaporeSingaporeSingapore
  2. 2.NUS Graduate School for Integrative Sciences and Engineering (NGS), National University of SingaporeSingaporeSingapore

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