  We focus on the special situation of $D=2J$ of the general spin-S Blume-Capel
model on the square lattice. Under the infinitesimal external magnetic field,
the phase transition behaviors due to the thermal fluctuations are discussed by
the newly developed tensor renormalization group method. For the case of the
integer spin-S, the system will undergo $S$ first-order phase transitions with
the successive symmetry breaking with the magnetization $M=S,S-1,...0$. For the
half-integer spin-S, there are similar $S-1/2$ first order phase transition
with $M=S,S-1,...1/2$ stepwise structure, in addition, there is a continuous
phase transition due to the spin-flip $Z_2$ symmetry breaking. In the low
temperature regions, all first-order phase transitions are accompanied by the
successive disappearance of the optional spin-component pairs($s,-s$),
furthermore, the critical temperature for the nth first-order phase transition
is the same, independent of the value of the spin-S. In the absence of the
magnetic field, the visualization parameter characterizing the intrinsic
degeneracy of the different phases clearly demonstrates the phase transition
process.
