Numerical analysis of a leapfrog ADI-FDTD method for Maxwell's equations in lossy media

Recently, a so-called one-step leapfrog ADI-FDTD method has been developed in engineering community for solving the 3D time-dependent Maxwell's equations. This method becomes quite popular in simulation wave propagation in graphene-based devices due to its efficiency. We investigate this method from a theoretical point of view by proving the energy conservation property, the unconditional stability of this ADI-FDTD method, and establishing the optimal second-order convergence rate in both time and space on non-uniform cubic grids. Numerical results are presented justifying our analysis.