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Interesting applications of metamaterials have been proposed in the recent past, and in several cases the use of double-negative (DNG) slabs is appealing. Consequently, a detailed analysis of the propagation of electromagnetic fields inside DNG slabs is critical, and suitable numerical methods are needed. To this aim, a state-of-the-art FDTD approach is here proposed. It is based on a variable-mesh strategy, on the top of a parallel design, compatible with Grid computing technologies. It embeds the Drude model, and a wide variety of sources. Results demonstrate the accuracy and high-performance of the tool, and propose a detailed characterization of DNG slabs.