Rigorous analysis of the time-splitting methods for the semiclassical Dirac equation

Abstract

We provide rigorous error analysis of the mass-preserving time-splitting methods for solving the semiclassical Dirac equation. The scaled Planck constant $\hat{I}\mu$ in the equation gives rise to rapid oscillations in both space and time when $0<\hat{I}\mu \ll 1$ with wavelengths of order $O(\hat{I}\mu )$. %We prove that the first-order splitting $S_1$ and the second-order splitting $S_2$ schemes preserve the total discretized mass. Rigorous error estimates reveal the precise dependence of the approximation errors on the time step $\ddot{I}„$, the spatial mesh size $h$, and the parameter $\hat{I}\mu$. Specifically, the temporal error scales as $O\left(\ddot{I}„/\hat{I}{\mu }^2\right)$ for the first-order splitting $S_1$ and as $O\left(\ddot{I}{„}^2/\hat{I}{\mu }^3\right)$ for the second-order splitting $S_2$, while the spatial error scales as $O(h^m/\hat{I}{\mu }^m)$ for both methods, where $m$ is related to the regularity of the solution. In addition, we obtain error bounds for key physical observables, including the total probability density $\ddot{I}$ and the current density $\mathbf{J}$. Compared with finite difference time domain (FDTD) methods, time-splitting approaches exhibit spectral accuracy in space and retain a relatively low computational cost. Furthermore, we demonstrate that higher accuracy can be achieved by employing the fourth-order compact time-splitting ($S_{\text{4c}}$) method. Numerical experiments are conducted to verify the reliability of the error estimates.

22 pages, 1 figure