Gaussian Beam Methods for the Dirac Equation in the Semi-classical Regime

TL;DR

This paper introduces an Eulerian Gaussian beam method for efficiently approximating solutions to the Dirac equation in the semi-classical regime, reducing computational cost while maintaining accuracy.

Contribution

It develops a novel Eulerian Gaussian beam approach that leverages eigenvalue decomposition for independent eigenspace evolution, improving efficiency over traditional methods.

Findings

The method achieves high accuracy in numerical experiments.

It significantly reduces computational cost compared to existing techniques.

The approach effectively handles the semi-classical Dirac equation.

Abstract

The Dirac equation is an important model in relativistic quantum mechanics. In the semi-classical regime $\epsilon\ll1$, even a spatially spectrally accurate time splitting method \cite{HuJi:05} requires the mesh size to be $O(\epsilon)$, which makes the direct simulation extremely expensive. In this paper, we present the Gaussian beam method for the Dirac equation. With the help of an eigenvalue decomposition, the Gaussian beams can be independently evolved along each eigenspace and summed to construct an approximate solution of the Dirac equation. Moreover, the proposed Eulerian Gaussian beam keeps the advantages of constructing the Hessian matrices by simply using level set functions' derivatives. Finally, several numerical examples show the efficiency and accuracy of the method.