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.

中文翻译：

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Dirac方程的四阶紧致时分傅里叶拟谱方法

我们为Dirac方程提出了一种新的四阶紧致时间分裂（\（S_ \ mathrm {4c} \））傅里叶伪谱方法，方法是将Dirac方程分为两部分，并使用它们之间的双换向器对Dirac进行积分每个时间间隔的方程。该方法是显式的，在时间上是四阶的，在空间上是频谱的。它是无条件稳定的，并且在离散级上保留了总概率。之所以称为紧凑时间分割方法，是因为在每个时间步，\（S_ \ mathrm {4c} \）中的子步数比标准的四阶分割方法和四阶分割方法少得多。龙格-库塔分裂方法。\（S_ \ mathrm {4c} \）的另一个优点这样做是为了避免在每个时间间隔对子问题进行积分时使用负的时间步长。从精度，效率和长时间行为的角度，比较了\（S_ \ mathrm {4c} \）与其他许多现有的Dirac方程时间分割方法。数值结果证明了所提出的\（S_ \ mathrm {4c} \）在效率和准确性方面的优势。最后，我们报告了Dirac方程在不同参数范围内的\（S_ \ mathrm {4c} \）的时空分辨率，包括非相对论的极限范围，半经典的极限范围以及同时的非相对论和无质量的极限范围。