  With Monte Carlo methods, we simulate the critical domain-wall dynamics of
model B, taking the two-dimensional Ising model as an example. In the
macroscopic short-time regime, a dynamic scaling form is revealed. Due to the
existence of the quasi-random walkers, the magnetization shows intrinsic
dependence on the lattice size $L$. A new exponent which governs the
$L$-dependence of the magnetization is measured to be $\sigma=0.243(8)$.
