Numerical Analysis of a Leapfrog ADI-FDTD Method for Metamaterial Maxwell's Equations

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 simulating wave propagation in graphene-based devices due to its computational efficiency. We extend this ADI method to solve metamaterial Maxwell's equations and investigate it from a theoretical point of view by proving its energy conservation property, the unconditional stability, 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.