A time-splitting spectral scheme for the Maxwell-Dirac system

TL;DR

This paper introduces an efficient, stable, and gauge-conserving spectral scheme for simulating the Maxwell-Dirac system, capable of handling semi-classical and non-relativistic regimes with uniform convergence.

Contribution

A novel time-splitting spectral method for the Maxwell-Dirac system that preserves gauge conditions and demonstrates uniform convergence across regimes.

Findings

Conserves Lorentz gauge condition.

Unconditionally stable and highly efficient.

Shows uniform convergence in the non-relativistic limit.

Abstract

We present a time-splitting spectral scheme for the Maxwell-Dirac system and similar time-splitting methods for the corresponding asymptotic problems in the semi-classical and the non-relativistic regimes. The scheme for the Maxwell-Dirac system conserves the Lorentz gauge condition, is unconditionally stable and highly efficient as our numerical examples show. In particular we focus in our examples on the creation of positronic modes in the semi-classical regime and on the electron-positron interaction in the non-relativistic regime. Furthermore, in the non-relativistic regime, our numerical method exhibits uniform convergence in the small parameter $\dt$, which is the ratio of the characteristic speed and the speed of light.