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 $\epsilon$ in the equation gives rise to rapid oscillations in both space and time when $0<\epsilon\ll 1$ with wavelengths of order $O(\epsilon)$. %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 $\tau$, the spatial mesh size $h$, and the parameter $\epsilon$. Specifically, the temporal error scales as $O\left(\tau/\epsilon^2\right)$ for the first-order splitting $S_1$ and as $O\left(\tau^2/\epsilon^3\right)$ for the second-order splitting $S_2$, while the spatial error scales as $O(h^m/\epsilon^m)$ 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 $\rho$ 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.