Energy conservation and super convergence analysis of the EC-S-FDTD schemes for Maxwell equations with periodic boundaries

This paper is concerned with new energy analysis of the two dimensional Maxwell's equations and the symmetric energy-conserved splitting finite difference time domain (EC-S-FDTD) method with the periodic boundary (PB) condition. New energy identities of the Maxwell's equations in terms of H-1 and H-2 norms are proposed and interpreted by considering the physical meanings of the H-1 and H-2 semi-norms in the identities. It is found from these new identities that the first and second curls of the electromagnetic fields are conserved in terms their magnitudes. By the energy methods, the numerical energy identities of the symmetric EC-S-FDTD method are derived and shown to converge to the continuous energy identities of the Maxwell's equations. This proves that the symmetric EC-S-FDTD scheme is unconditionally stable and energy conserved in the discrete H-1 and H-2 norms. Also by the energy methods, it is proved that the symmetric EC-S-FDTD method with PB condition is of second order (super) convergence in the discrete H-1 and H-2 norms. Numerical experiments are carried out and confirm the analysis on energy conservation, stability and super convergence.