we can conveniently calculate thermodynamic variables such as $C_{\rm p}$, $\nabla_{\rm a}$, $(\partial
T/\partial \rho)_e$, etc. Comparisons show that the discrepancies between the $\rho$, $C_{\rm p}$ and $\nabla_{\rm a}$ calculated by equation (\[eos\]) and the interpolation from the OPAL EOS tables ([@rogers96]) are less than $2\%$ in the regions we computed.

The opacity $\kappa$ is obtained by interpolating the OPAL tables ([@rogers92]). Before interpolation, the OPAL opacity is tabulated as functions of density and $\mathcal{R}=\rho/(T\times
10^{-6})^3$ in two-dimensional arrays with even $\Delta \ln{\rho}$ and $\Delta \ln{\mathcal{R}}$. A *hunt* method is used to search the table.

Initial Models: Construction and Modifications
----------------------------------------------

   $M/M_\odot$   $\log{(T_e)}$   $\log{(L/L_\odot)}$       Age($yrs$)       $R/R_\odot$
  ------------- --------------- --------------------- -------------------- -------------
        3            3.695              2.032          $4.258\times 10^8$     14.113
       15            3.761              4.869          $1.308\times 10^7$     273.376

  : Key properties of the reference stars.[]{data-label="key"}

Both of these two stars have the same chemical composition : $X=0.7, Z=0.02$.

We consider two types of post-main-sequence stars: a massive star of 15$M_\odot$ (red super giant) and an intermediate mass star of 3$M_\odot$ (red giant). The regions we study are far away from the core and surface, so that we need not consider the Non-LTE (local thermal equilibrium) effects and nuclear reaction. Our 1D reference stellar models are 