A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials

TL;DR

This paper introduces a fourth-order compact time-splitting method for solving the Dirac equation with time-dependent electromagnetic potentials, improving accuracy and efficiency while maintaining ease of implementation.

Contribution

The paper develops a novel fourth-order compact time-splitting scheme incorporating time-ordering for the Dirac equation with time-dependent potentials.

Findings

The scheme achieves high accuracy in 1D and 2D numerical tests.

It efficiently handles time-dependent potentials with limited modifications.

Numerical results validate the method's effectiveness and simplicity.

Abstract

In this paper, we present an approach to deal with the dynamics of the Dirac equation with time-dependent electromagnetic potentials using the fourth-order compact time-splitting method ($S_\text{4c}$). To this purpose, the time-ordering technique for time-dependent Hamiltonians is introduced, so that the influence of the time-dependence could be limited to certain steps which are easy to treat. Actually, in the case of the Dirac equation, it turns out that only those steps involving potentials need to be amended, and the scheme remains efficient, accurate, as well as easy to implement. Numerical examples in 1D and 2D are given to validate the scheme.