Stability and Super Convergence Analysis of ADI-FDTD for the 2D Maxwell Equations in a Lossy Medium

Several new energy identities of the two dimensional(2D) Maxwell equations in a lossy medium in the case of the perfectly electric conducting boundary conditions are proposed and proved. These identities show a new kind of energy conservation in the Maxwell system and provide a new energy method to analyze the alternating direction implicit finite difference time domain method for the 2D Maxwell equations (2D-ADI-FDTD). It is proved that 2D-ADI-FDTD is approximately energy conserved, unconditionally stable and second order convergent in the discrete L2 and H1 norms, which implies that 2D-ADI-FDTD is super convergent. By this super convergence, it is simply proved that the error of the divergence of the solution of 2D-ADI-FDTD is second order accurate. It is also proved that the difference scheme of 2D-ADI-FDTD with respect to time t is second order convergent in the discrete H1 norm. Experimental results to confirm the theoretical analysis on stability, convergence and energy conservation are presented.