The utilization of a fourth-order Modified Runge-Kutta (MRK) scheme on a transformed coordinate system for simulating electromagnetic applications that involve non-rectangular domains are presented. The model equations were obtained by writing the governing Maxwell’s equations in the curvilinear coordinate system. Therefore, by transforming the arbitrary-shaped structures to a uniform rectangular grid, numerical schemes and boundary conditions can be easily implemented. The time advancement of fields in the computational space is performed by leapfrogging the stages of explicit fourth-order MRK method. The one-dimensional traveling wave simulation results indicate that the leapfrogging of MRK stages reduces the dissipation errors associated with the MRK solution of wave propagation. The computer algorithms based on this technique have been developed for near-field and far-field scattering applications. Several classes of problems have been investigated in the dissertation. In the initial stage, accuracy of the scheme has been established by simulating two standing wave problems and comparing MRK results with the existing exact solution. The results for both cases are found to be in good agreement. In the second stage, the total field-scattered field (TF-SF) formulation is incorporated in the algorithm for the simulation of far-field scattering. The two-dimensional and three-dimensional computer codes developed based on this algorithm are used to perform qualitative analysis using planes waves and cylindrical waves as source. The diffraction and penetration of the waves are studied in the presence of dielectric or conducting materials having linear, non-dispersive, and homogeneous properties. The results are compared with the results from the benchmark method and indicates good agreement. The grid independence investigations revealed the dependence on the Courant number to obtain accurate results. Subsequently, scattering investigations are carried out on non-rectangular geometries with the clustered grid points. Finally, computer algorithm is modified to include the modeling of dispersive (frequency-dependent) material, and the absorption and reflection of electromagnetic waves by biological tissues is studied in one-dimension. The predicted reflection coefficient for two separate studies indicate excellent agreement with the exact reflection coefficients for a wideband frequency range.