Frozen Gaussian Approximation for the Dirac Equation in Semiclassical Regime

Abstract

This paper focuses on the derivation and analysis of the frozen Gaussian approximation (FGA) for the Dirac equation in the semiclassical regime. Unlike the strictly hyperbolic system studied in [J. Lu and X. Yang, Comm. Pure Appl. Math., 65 (2012), pp. 759--789], the Dirac equation possesses eigenfunction spaces of multiplicity two, which demands more delicate expansions for deriving the amplitude equations in FGA. Moreover, we prove that the nonrelativistic limit of the FGA for the Dirac equation is the FGA of the Schrödinger equation, which shows that the nonrelativistic limit is asymptotically preserved after one applies FGA as the semiclassical approximation. Numerical experiments including the Klein paradox are presented to illustrate the method and confirm part of the analytical results.