Dynamics of spontaneous capillary penetration of a liquid into a cylindrical pore is studied numerically over the entire duration of an experiment, including the initial stages of penetration during which inertial effects are dominant. Partial slip in the vicinity of the moving contact-line is allowed by using an empirical constitutive relation between the dynamic contact angle and contact line speed in order to avoid the stress singularity arising from the presence of the moving contact line on the solid wall. A finite-difference scheme on a staggered body-fitted grid is used to solve for the time-dependent flow field and to determine the time evolution of the shape of the advancing meniscus. The results of dynamic simulations of capillary rise under both normal and microgravity conditions are compared with the reported experimental observations. The simulation results are found to be in good agreement with the experimental measurements in both cases. Numerical simulations capture the different flow regimes identified in previous studies of spontaneous capillary penetration. The velocity dependence of the dynamic contact line is found to have a significant effect on kinetics of wetting in the intermediate-time flow regime, wherein the capillary force is balanced by convective losses.