Numerical calculations of spatio-temporal characteristics of the breakdown in the dielectric barrier discharge in neon at atmospheric pressure between parallel-plate electrodes were performed using one-dimensional fluid model. The hydrodynamic set of equations (continuity equations with Poisson equation) was solved using the improved Sharfetter-Gummel algorithm combined with the adaptive step Runge{Kutta method. This algorithm allowed the simulation with high gradient of particle density in reasonable time. The time development of ion and electron densities during the avalanche phase and cathode streamer propagation phase was presented. The influence of the parameters of dielectrics on the breakdown was analyzed.