In this paper, the construction of time-dependent solution of electromagnetic fields in space-time by exploiting Taylor Series and Geometrical Optics' (GO) tools, i.e. wavefronts and rays, is introduced. Discontinuities in the fields and their successive time derivatives may only exist on the wavefronts and propagate along the rays. These discontinuities are transported via higher order transport equations. The proposed method is implemented by two different procedures on a sample problem of Hertzian dipole at origin in isotropic, homogeneous medium. In the first one, discontinuities themselves are transported directly by differential-type equations but in the latter by auxiliary vectors using both differential and integral type equations and conservation of energy is taken into account. Forward differences in time and central differences in space are applied. Simulation studies, when compared with analytical results, show that consistent and accurate results are obtained.