star models are constructed with different values of $a_{\rm f}$ (see Table \[runs\]). If $a_{\rm f}$ is much greater than that of Case I3, the program can easily crash, and a value smaller than that of Case I1 would make the results affected by the numerical errors (truncation and round-off errors). In Table [\[runs\]]{} $F_{\rm b}$ is the modified dimensionless numerical input energy flux, i.e., $F_{\rm b}=F_{\rm btm}c_{\rm r}a_{\rm f}$. The initial superadiabatic gradient and radiation flux for the intermediate mass star models are shown in Fig. \[init\]. The three models have the same initial $\Delta\nabla$ (see the solid line in bottom panel of Fig. \[init\]). Their $F_{\rm rad}$ have the same shape but different magnitudes, the top panel of Fig. \[init\] just gives an example of radiation flux distribution. Two times the initial $F_{\rm rad}$ of Case I3 is plotted for clarity. Note that the constructed intermediate mass star models are based on local convection theory, i.e., there is no overshoot hump in the radiative flux distribution (compare with Fig.5 of [@xiong01], hereafter XD2001).

  Identifier     $a_{\rm f}$    $F_{\rm b}$     $t$     $\sigma_{\rm max}$
  ------------ ---------------- ------------- ------- --------------------
  I1            $1\times 10^4$  0.05           9295                   5.9%
  I2            $2\times 10^4$  0.10           7316                   1.7%
  