Mathematical and Numerical Aspects of Quantum Dynamics

A micro-macro numercial method solving a multi-scale model for graphene

Mohammed Lemou

CNRS and University of Rennes 1, France

A class of kinetic graphene models has been recently derived within a semiclassical limit where important interband quantum transitions are taken into account. We propose a micro-macro decomposition based numerical approach which solves the highly oscillatory model in the original coordinate, yet can capture numerically the oscillatory space-time quantum solution pointwisely even without numerically resolving the frequency. This reduces the computational dimension of the nonlinear geometric optics method based numerical method that was recently introduced to solve highly oscillatory transport equations. We prove that the underlying micro-macro equations have smooth (up to certain order of derivatives) solutions with respect to the frequency, and then prove the uniform accuracy of the numerical discretization for a scalar model equation exhibiting the same oscillatory behavior. Numerical experiments verify the theory.