To improve the beam brightness produced by a Source of Negative Ions by Cesium Sputtering we studied the beam generation in the 12~MeV Vandergraff linear accelerator at Instituto Nacional de Investigaciones Nucleares. Results of 3D particle tracking simulations of the ion source and beamline have been compared with measurements, with better agreement than traditional codes that only take into account the negative beam, and they determine a suppression in the Cs$^{+}$ production due to space charge, which in turn explains the intensity limits for negative beam production in both ionizers, and the best way to overpass them. Also, the beam dynamics variation due to the erosion of the target inside the cathode has been determined, helping to prevent beam losses and enhance the beam brightness.