Error bounds of fourth‐order compact finite difference methods for the Dirac equation in the massless and nonrelativistic regime

We establish the error bounds of fourth-order compact finite difference (4cFD) methods for Dirac equation in massless and nonrelativistic regime, which involves a small dimensionless parameter $0 < \varepsilon \le 1$ inversely proportional to speed light. In this solution propagates waves with wavelength $O(\varepsilon)$ time $O(1)$ space, as well wave $O(1/\varepsilon)$ rapid outgoing waves. adapt conservative semi-implicit 4cFD discretize rigorously carry out their depending explicitly on mesh size $h$, step $\tau$ $\varepsilon$. Based bounds, $\varepsilon$-scalability is $h = O(\varepsilon^{1/4})$ $\tau O(\varepsilon^{3/2})$, not only improves spatial resolution capacity but also has superior accuracy than classical second-order methods. Furthermore, physical observables including total density current have same conclusions. Numerical results are provided validate dynamics different potentials 2D presented.