This article presents the results of numerical investigations of initial stages of a fast capillary discharge-initiation and propagation of a fast ionization wave and consequent formation of a conducting plasma channel. Simulations were performed for the case of a 10 cm-long narrow capillary filled with argon at 270 Pa. The wave was created by voltage pulses of negative polarity with an amplitude of 10 kV and rise times varying in the 5-40 ns range, which is comparable with switching times of conventional switching devices based on gas discharges. Obtained spatio-temporal distributions of main discharge parameters were used to study the wave dynamics in general and the structure of the wavefront and the plasma channel. Simulations have revealed that the total wave propagation time is comparable to the voltage rise-time and varies linearly from 10 to 30 ns for the considered rise-time range. It is further shown that for longer voltage rise-times, it is possible for the wavefront to reach the anode well before the applied voltage reaches the maximum, which can potentially result in suboptimal initiation of a fast capillary discharge with lower current rise rates and less effective energy input.