Frozen Gaussian Approximation for the Dirac equation in Semi-classical regime

      In this talk we introduce the derivation and analysis of the Frozen Gaussian Approximation (FGA) for the Dirac equation in the semi-classical regime. Unlike the strictly hyperbolic system studied in [Lu&Yang CPAM2012], 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. This is a joint work with Emmanuel Lorin and Xu Yang