computed with more realistic input physics and Xiong’s non-local time dependent stellar turbulent convection theory (private communication). The model properties (i.e., mass $M$, effective temperature $T_e$, luminosity $L$, age and radius $R$) are listed in Table \[key\].

In the simulation of the intermediate mass star envelope, we have encountered serious difficulties. They are discussed as follows.

1.  The first problem is the very long duration of thermal relaxation. The thermal relaxation time scale can be estimated by dividing the total energy contained in the system by the rate of energy input at the bottom. For intermediate mass star, the dimensionless input energy flux ($F_{\rm b}$) is of the order of $10^{-5}$ and the radial integral of energy density energy ($e$) above the bottom ($z=0$) is on the order of $1000$. This means that the system needs a time scale of $10^8$ to reach a thermally relaxed state. The time step achieved by the ADISM scheme is around $10^{-2}$ for current study. Therefore, we need about $10^{10}$ steps to accomplish the relaxation. This requires too much computational resources.

2.  Meanwhile, the limited spatial resolution is unable to handle the small superadiabatic gradient ($\Delta\nabla={\partial\ln{T}}/{\partial\ln{p}}-\nabla_{\rm a}$) accurately. In the efficient convective region, $\Delta\nabla$ 