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The finite-difference time-domain (FDTD) method is applied for modelling of wire media as artificial dielectrics. Both frequency dispersion and spatial dispersion effects in wire media are taken into account using the auxiliary differential equation method. According to the authors' knowledge, this is the first time when the spatial dispersion effect is considered in the FDTD modelling. The stability of developed spatially dispersive FDTD formulations is analyzed through the use of von Neumann method combined with the Routh-Hurwitz criterion. The results show that the conventional stability Courant limit is preserved using standard discretization scheme for wire media modelling. Flat sub-wavelength lenses formed by wire media are chosen for validation of proposed spatially dispersive FDTD formulation. Results of the simulations demonstrate excellent sub-wavelength imaging capability of the wire medium slabs. The size of the simulation domain is significantly reduced using the modified perfectly matched layer (MPML) which can be placed in close vicinity of the wire medium. It is demonstrated that the reflections from the MPML-wire medium interface are less than-70 dB, that lead to dramatic improvement of convergence compared to conventional simulations.