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

We propose a new fourth-order compact time-splitting ($S_\text{4c}$) 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 unconditional stable and conserves the total density in the discretized level. It is called a compact time-splitting method since, at each time step, the number of sub-steps in $S_\text{4c}$ is much less than those of the standard fourth-order splitting method and the fourth-order partitioned Runge-Kutta splitting method. Comparison among $S_\text{4c}$ and many other existing time-splitting methods for the Dirac equation are carried out in terms of accuracy and efficiency as well as long time behavior. Numerical results demonstrate the advantage in terms of efficiency and accuracy of the proposed $S_\text{4c}$. Finally we report the spatial/temporal resolutions of $S_\text{4c}$ for the Dirac equation in different parameter regimes including the nonrelativistic limit regime, the semiclassical limit regime, and the simultaneously nonrelativisic and massless limit regime.