The numerical solver for the velocity field equation describing laminar pulsatile flows driven by a time-dependent pressure drop in the straight microfluidic channel of square cross-section is developed. In the computational algorithm, an orthogonal collocation on finite element scheme for spatial discretizations is combined with an adaptive Runge-Kutta method for time integration. The algorithm with the 1,521 computational nodes and the accuracy up to O($10^{-5}$) is applied to the flow in the back-and-forth standing mode with the channel hydraulic diameter ($D_h$) in the range 10-500 ${mu}m$ and the oscillating frequency ($f$) of 1 to 100 Hz. As a result, a periodic steady state is defined as the flow condition where there would be no net movement after long time elapses. Following by the retardation phenomena in a cycle, reversal of the axial velocity is observed at the channel center. Major attention is focused on the influences of the size of channel cross-section and the oscillating frequency. Increasing $D_h$ and $f$ results in the decrease in the amplitude of mean velocity but the increase in the start-up time. Larger time delay occurs by low-frequency pulsation.