Super-resolution of time-splitting methods for the Dirac equation in the nonrelativistic regime

Abstract:

We establish error bounds of the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) for the Dirac equation in the nonrelativistic regime in the absence of external magnetic potentials, with a small parameter $0<\varepsilon \leq 1$ inversely proportional to the speed of light. In this regime, the solution propagates waves with $O(\varepsilon ^2)$ wavelength in time. Surprisingly, we find out that the splitting methods exhibit super-resolution, i.e., the methods can capture the solutions accurately even if the time step size $\tau$ is independent of $\varepsilon$, while the wavelength in time is at $O(\varepsilon ^2)$. $S_1$ shows $1/2$ order convergence uniformly with respect to $\varepsilon$, by establishing that there are two independent error bounds $\tau + \varepsilon$ and $\tau + \tau /\varepsilon$. Moreover, if $\tau$ is nonresonant, i.e., $\tau$ is away from a certain region determined by $\varepsilon$, $S_1$ would yield an improved uniform first order $O(\tau )$ error bound. In addition, we show $S_2$ is uniformly convergent with $1/2$ order rate for general time step size $\tau$ and uniformly convergent with $3/2$ order rate for nonresonant time step size. Finally, numerical examples are reported to validate our findings.