In this paper a novel computational approach simulating the transient Elasto-Hydrodynamic Lubrication of line-contacts is developed. It is based on the description of the fluid dynamics, solid mechanics and load dynamics occurring simultaneously in the writing of a unified system of Differential Algebraic Equations, while they are discretised, respectively, by the means of the Finite Difference Method, the Boundary Element Method (accounting for the Fast Fourier Transform to evaluate the profiles' deformation) and the trapezoidal integration of the lubricant pressure. The assembled system is solved through the time marching executed by the means of Implicit Runge-Kutta methods feasible with the problem: three simulations were presented in order to analyse the stationary minimum film thickness after the sliding and load perturbations and to compare it with the well-established empirical one estimated by Dowson obtaining satisfactory matching results, while other three simulation were proposed to analyse pure squeeze, harmonic sliding and harmonic squeeze configurations, providing interesting discussions about the wave dynamics occurring in the lubricant meatus due to the profiles' approach dynamics. Furthermore, a selection between the used Runge-Kutta methods was provided based on their converging results, highlighting the robustness of the diagonal ones for this type of problems.