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In this paper, a complex nonstandard finite-difference time-domain (CNS-FDTD) method is proposed in order to simulate wave propagation in lossy media. To clarify the characteristics of the method, expressions for the numerical propagation constant and the stability condition are derived. It is found that the CNS-FDTD method is much more accurate and stable than the conventional finite-difference time-domain (FDTD) method. The method is applied to the analysis of a fin ferrite electromagnetic wave absorber with a periodic structure.