With the rapid increases in computing power, the use of numerical simulation method of gas discharge process has become a hot spot. Numerical oscillation and numerical diffusion issues exist in the solving hydrodynamic models derived from gas discharge theory, which belong to convection-dominant equation. In this paper, Galerkin weak form of weighted residual method is used to solve the model, and streamer discharge process in a 10mm rod-plate air gap is simulated. Simulation results show that: the electron, positive ion’s concentration in the streamer discharge process both are 1016cm-3 orders of magnitude, negative ions is 1015cm-3；electron impact ionization leads role, and the space charge has obvious distortion on electric field distribution; the numerical oscillation and numerical diffusion have been effectively eliminated.