In this paper, we consider a deterministic approach for modeling angiogenesis. The model equations form a nonlinear coupled system of partial and ordinary differential equations. We propose an efficient, accurate and locally conservative numerical method to solve the nonlinear system. Computational results indicate that the model generates the overall dendritic structure and pattern of the capillary network morphologically similar to those observed in vivo. The influence of the capillary network and the growth factor distribution on each other and their interaction are investigated using numerical simulations.