  We investigate the tricritical scaling behavior of the two-dimensional
spin-$1$ Blume-Capel model using the Wang-Landau method measuring the joint
density of states for lattice sizes up to $48\times 48$ sites. The first-order
transition curve is systematically determined employing the method of field
mixing in conjunction with finite-size scaling, showing a significant deviation
from the previous data points. Deep in the first-order area of the phase
diagram, we also find that the specific heat exhibits a double-peak structure
of the Schottky-like anomaly appearing with the transition peak. At the
tricritical point, we characterize the tricritical exponents through
finite-size scaling analysis including the phenomenological finite-size scaling
with thermodynamic variables. Our estimation of the tricritical eigenvalue
exponents, $y_t = 1.804(5)$, $y_g = 0.80(1)$, and $y_h = 1.925(3)$, provides
the first Wang-Landau verification of the conjectured exact values,
demonstrating the effectiveness of the density-of-states-based approach in
finite-size scaling study of multicritical phenomena.
