  Herein, we propose a site random cluster model by introducing an additional
cluster weight in the partition function of the traditional site percolation.
To simulate the model on a square lattice, we combine the color-assignation and
the Swendsen-Wang methods together to design a highly efficient cluster
algorithm with a small critical slowing-down phenomenon. To verify whether or
not it is consistent with the bond random cluster model, we measure several
quantities such as the wrapping probability $R_e$, the percolation strength
$P_\infty$, and the magnetic susceptibility per site $\chi_p$ as well as two
exponents such as the thermal exponent $y_t$ and the fractal dimension $y_h$ of
the largest cluster. We find that for different exponents of cluster weight
q=1.5, 2, 2.5, 3, 3.5 and 4, the numerical estimation of the exponents $y_t$
and $y_h$ are consistent with the theoretical values. The universalities of the
site random cluster model and the bond random cluster model are completely
identical. For larger values of $q$, we find obvious signatures of the
first-order percolation transition by the histograms and the hysteresis loops
of the percolation strength and the energy per site. Our results are helpful
for the understanding of the percolation of traditional statistical models.
