---
abstract: 'An attempt at understanding the downward overshooting in the convective envelopes of the post-main-sequence stars has been made on the basis of three-dimensional large eddy simulations, using artificially modified OPAL opacity and taking into account radiation and ionization in the equation of state. Two types of stars, an intermediate mass star and a massive star were considered. To avoid the long thermal relaxation time of the intermediate mass star, we increased the stellar energy flux artificially while trying to maintain a structure close to the one given by one-dimensional stellar model. A parametric study of the flux factor was performed. For the massive star, no such manner was necessary. Numerical results were analyzed when the system reached the statistical steady state. It was shown that the penetration distance in pressure scale heights is of the order of unit. The scaling relations among the penetration distance, input flux and vertical velocity fluctuations studied by Singh et al. (1998) were checked. The anisotropy of the turbulent convection and the diffusion models of third order moments representing the non-local transports were also investigated. These models are dramatically affected by the velocity fields and no universal constant parameters seem to exist. The 