  The Sweeny algorithm for the $Q$-state random-cluster model in two dimensions
is shown to exhibit a rich mixture of critical dynamical scaling behaviors. As
$Q$ decreases, the so-called critical speeding-up for non-local quantities
becomes more and more pronounced. However, for some quantity of specific local
pattern -- e.g., the number of half faces on the square lattice, we observe
that, as $Q \to 0$, the integrated autocorrelation time $\tau$ diverges as
$Q^{-\zeta}$, with $\zeta \simeq 1/2$, leading to the non-ergodicity of the
Sweeny method for $Q \to 0$. Such $Q$-dependent critical slowing-down,
attributed to the peculiar form of the critical bond weight $v=\sqrt{Q}$, can
be eliminated by a combination of the Sweeny and the Kawasaki algorithm.
Moreover, by classifying the occupied bonds into bridge bonds and backbone
bonds, and the empty bonds into internal-perimeter bonds and external-perimeter
bonds, one can formulate an improved version of the Sweeny-Kawasaki method such
that the autocorrelation time for any quantity is of order $O(1)$.
