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

TL;DR

This paper derives error bounds for fourth-order compact finite difference methods applied to the Dirac equation in a specific regime, showing improved accuracy and resolution over classical methods, with validation through numerical experiments.

Contribution

The paper establishes rigorous error bounds for 4cFD methods for the Dirac equation in the massless, nonrelativistic regime, revealing their superior accuracy and resolution capacity.

Findings

Error bounds depend explicitly on mesh size, time step, and small parameter.

$ ext{ε}$-scalability: $h=O( ext{ε}^{1/4})$, $ au=O( ext{ε}^{3/2})$.

Numerical results validate theoretical error bounds and dynamics.

Abstract

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