AbstractA brief review is given of a systematic, product‐formula based approach to construct unconditionally stable algorithms for solving the time‐dependent Maxwell equations. The fundamental difficulties that arise when we want to incorporate uniaxial perfectly matched layer boundary conditions into this scheme is discussed. We construct an algorithm that circumvents these difficulties and is unconditionally stable. Results of simulations for a point source and a plane current source inside a three‐dimensional volume illustrate that in practice, the algorithm performs as theoretically expected. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)