We develop an ab initio time-dependent Basis Function (tBF) method to solve non-perturbative and time-dependent problems in non-relativistic quantum mechanics. As a test problem, we apply this method to the Coulomb excitation of a deuteron by an impinging heavy ion. We employ wave functions for the bound and excited states of the deuterium system based on a realistic nucleon-nucleon interaction and study the evolution of the transition probability, the r.m.s. charge radius and the r.m.s. momentum of the system during the scattering process. The dependencies of these quantities on the external field strength and the bombarding energy are also analyzed and compared to corresponding results obtained from first-order perturbation theory. The time evolution of both the charge and the momentum distributions is shown to reveal the dynamics.