In order to improve many industrial applications of the plasma sheath and the cathode erosion of electrical arc, the prime objective of this paper is to treat the modelling and simulation of the effects of ions collisionality on the plasma sheath and to determine the collisional dependence of the sheath width and the ion impact energy on the wall using the Runge-Kutta Routrine on the presence or the absence of the term source in the continuity equation . Then , to explain the dependence of the measured erosion rate of the cathode by considering the cathode spot heat balance equation coupled with equation obtained from the sheath and taking into account the physical phenomena such as ion and electron fluxes, radiation, thermal conductivity and evaporation.