# Uniform error bounds of time-splitting methods for the nonlinear Dirac equation in the nonrelativistic regime without magnetic potential

## TL;DR

This paper rigorously analyzes super-resolution properties of Lie-Trotter and Strang splitting methods for the nonlinear Dirac equation in the nonrelativistic regime, revealing their uniform error bounds and super-resolution capabilities despite high oscillations.

## Contribution

It provides the first rigorous proof of super-resolution and uniform error bounds for splitting methods applied to the nonlinear Dirac equation without magnetic potential in the nonrelativistic limit.

## Key findings

- Both $S_1$ and $S_2$ exhibit 1/2 order convergence uniformly in $ ext{varepsilon}$.
- Non-resonant time steps improve convergence to first and 3/2 order for $S_1$ and $S_2$ respectively.
- Super-resolution holds for higher order splitting methods.

## Abstract

Super-resolution of the Lie-Trotter splitting ($S_1$) and Strang splitting ($S_2$) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter $0<\varepsilon\leq 1$ inversely proportional to the speed of light. In this regime, the solution highly oscillates in time with wavelength at $O(\varepsilon^2)$. The splitting methods surprisingly show super-resolution, i.e. the methods can capture the solution accurately even if the time step size $\tau$ is much larger than the sampled wavelength at $O(\varepsilon^2)$. Similar to the linear case, $S_1$ and $S_2$ both exhibit $1/2$ order convergence uniformly with respect to $\varepsilon$. Moreover, if $\tau$ is non-resonant, i.e. $\tau$ is away from certain region determined by $\varepsilon$, $S_1$ would yield an improved uniform first order $O(\tau)$ error bound, while $S_2$ would give improved uniform $3/2$ order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that super-resolution is still valid for higher order splitting methods.