In this paper we compare current implementations of commonly used numerical techniques - the Finite-Difference Time-Domain (FDTD) method, the Finite-Integration Technique (FIT), and Time-Domain Integral Equations (TDIE) - to solve the canonical problem of a horizontal dipole antenna radiating over lossless and lossy half-spaces. These types of environment are important starting points for simulating many Ground Penetrating Radar (GPR applications which operate in the near-field of the antenna, where the interaction among the antenna, the ground, and targets is important. We analysed the simulated current at the centre of the dipole antenna, as well as the electric field at different distances from the centre of the antenna inside the half-space. We observed that the results from the simulations using the FDTD and FIT methods agreed well with each other in all of the environments. Comparisons of the electric field showed that the TDIE technique agreed with the FDTD and FIT methods when observation distances were towards the far-field of the antenna but degraded closer to the antenna. These results provide evidence necessary to develop a hybridisation of current implementations of the FDTD and TDIE methods to capitalise on the strengths of each technique.