In this thesis, the alternating-direction implicit method (ADI) is investigated in conjunction with the finite difference time-domain method (FDTD) to allow crossing of the Courant-Friedrich-Levy (CFL) stability criterion while maintaining stability in the FDTD algorithm. The main reason for this is to be able to use a larger numerical time step than that governed by the CFL criterion. The desired effect is a significant reduction in numerical run-times. Although the ADI-FDTD method has been used in the literature, most analysis and application have been performed on simple three-dimensional cavities.This work makes original contribution in two aspects. Firstly, a new modified alternating-direction implicit method for a three-dimensional FDTD algorithm has been successfully developed and implemented in this research. This new method allows correct modelling of a realistic physical structure such as a microstrip patch with the ADI scheme without causing instability even when the CFL criterion is not observed. However, due to the inherent property of this modified ADI-FDTD method, a decreasing reflection coefficient is observed using this scheme.The second and more important contribution this research makes in the field of numerical electromagnetics is the development of a new method of simulating realistic complex structures such as geometries comprising copper patch antennas on a dielectric substrate. With this new method, for the first time, the ADl-FDTD algorithm remains stable while still in violation of the CFL criterion, even when complex structures are being modelled.However, there is a trade-off between accuracy and computational speed in ADI-FDTD and modified ADI-FDTD methods. The larger the numerical time step, the shorter is the simulation run-time but an increase in numerical time step causes a degradation in accuracy of numerical results. Comparison between speed and accuracy is shown in this thesis and it has to be mentioned here that these values are very much dependent on the structure being modelled
