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

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