Research on the droplet impact on a hydrophobic surface is of important theoretical significance and engineering value, both in mesoscopic fluid mechanics and interactions between microfluid and special materials. The van der Waals (vdW) equation of state relates the pressure to the temperature and the density of the fluid, and gives the long-range attractive force and short-range repulsive force between particles. The van der Waals equation of state can be used to describe the surface tension between liquid and vapor. As a pure meshless particle method, the smoothed particle hydrodynamic (SPH) method can use the vdW equation of state written in SPH form of N-S equations to describe the surface tension. The vdW surface tension mode is validated by simulating the coalescence of two equally sized static droplets in vacuum. Repellant of the hydrophobic surface is derived from a core potential. By combining the vdW surface tension and the repulsive force of the surface, the phenomenon of a liquid droplet impact with a certain initial velocity on the hydrophobic surface is studied. The SPH model is not only capable to describe the spreading of the droplet after it contacts the surface, but also clearly reproduces the springback, bouncing and secondary impact of the droplet. During the deformation of the droplet, the inertia force impels the spreading process of the droplet whilst the springback and bouncing behavior is dominated by the surface tension. The simulated results are in good agreement with the published experimental observations and VOF simulated results, indicating that the way we treat the surface tension and the repulsive force of the hydrophobic surface is effective and applicable in droplet impact surface problems. The impact velocity and liquid viscosity are considered to be two important factors that affect the deformation of the droplet after it contacts the surface. By varying the impact velocity within a certain range it is concluded that the maximum liquid-solid contact area increases as the impact velocity grows, and the bounced droplet will leave the surface when the velocity is big enough. Another comparison between different liquid viscosities shows that the maximum contact area decreases as the liquid viscosity increases because of the viscous dissipation, and the droplet barely rebound when the liquid viscosity is big enough.