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This paper is concerned with the optimal error estimates and energyconservation properties of the alternating direction implicit finite-differencetime-domain (ADI-FDTD) method which is a popular scheme for solving the 3DMaxwell equations. Precisely, for the case with a perfectly electric conducting(PEC) boundary condition we establish the optimal second-order error estimatesin both space and time in the discrete $H^1$-norm for the ADI-FDTD scheme andprove the approximate divergence preserving property that if the divergence ofthe initial electric and magnetic fields are zero then the discrete $L^2$-normof the discrete divergence of the ADI-FDTD solution is approximately zero withthe second-order accuracy in both space and time. A key ingredient is two newdiscrete energy norms which are second-order in time perturbations of two newenergy conservation laws for the Maxwell equations introduced in this paper.Furthermore, we prove that, in addition to two known discrete energy identitieswhich are second-order in time perturbations of two known energy conservationlaws, the ADI-FDTD scheme also satisfies two new discrete energy identitieswhich are second-order in time perturbations of the two new energy conservationlaws. This means that the ADI-FDTD scheme is unconditionally stable under thefour discrete energy norms. Experimental results are presented which confirmthe theoretical results.

Optimal error estimates and energy conservation identities of the ADI-FDTD scheme on staggered grids for 3D Maxwell's equations